Differential Equation Solver & Calculator
Accurately solve and visualize differential equations. Understand the underlying mathematics and their real-world applications.
Differential Equation Calculator
Linear First-Order ODE
Separable ODE
Second-Order Linear ODE (Constant Coeff)
Select the type of differential equation you want to solve.
Calculation Results
Enter values to see results.
Formula Used:
Select an equation type to see the relevant formula.
| Parameter | Value | Unit |
|---|
What is a Differential Equation?
A differential equation is a mathematical equation that relates a function with its derivatives. In simpler terms, it describes how a quantity changes. These equations are fundamental in modeling various phenomena across science, engineering, economics, and biology. They are used to describe everything from the motion of planets and the spread of diseases to the flow of heat and the growth of populations. Understanding differential equations is crucial for anyone looking to model dynamic systems and predict their behavior over time.
Who should use it: This calculator and the underlying concepts are essential for students of mathematics, physics, engineering, computer science, economics, and biology. Researchers and professionals in these fields use differential equations daily to model and analyze complex systems. Anyone interested in understanding how systems evolve based on their current state and rates of change will find value in exploring differential equations.
Common misconceptions: A frequent misconception is that differential equations are purely abstract mathematical constructs with no real-world relevance. In reality, they are the language used to describe many physical laws and natural processes. Another misconception is that solving differential equations is always incredibly complex and requires advanced computational power. While some can be challenging, many common types have well-established analytical or numerical solution methods, often solvable with tools like this calculator.
Differential Equation Solver Formula and Mathematical Explanation
The specific formula and method used depend heavily on the type of differential equation. This calculator supports several common types, each with its own solution technique.
Linear First-Order ODE: dy/dx + P(x)y = Q(x)
This is a fundamental type of differential equation. The general form is $ \frac{dy}{dx} + P(x)y = Q(x) $. The solution involves an integrating factor, $ \mu(x) = e^{\int P(x) dx} $. Multiplying the entire equation by the integrating factor transforms the left side into the derivative of a product: $ \frac{d}{dx}(\mu(x)y) = \mu(x)Q(x) $. Integrating both sides gives $ \mu(x)y = \int \mu(x)Q(x) dx + C $, where C is the constant of integration. Finally, solving for y yields the general solution: $ y(x) = \frac{1}{\mu(x)} \left( \int \mu(x)Q(x) dx + C \right) $.
Variables:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| P(x) | Coefficient of y (function of x) | Depends on context (e.g., 1/time) | Real numbers |
| Q(x) | Non-homogeneous term (function of x) | Depends on context (e.g., rate) | Real numbers |
| C | Constant of Integration | N/A | Real numbers |
| x | Independent variable | e.g., time, distance | Real numbers |
| y(x) | Dependent variable (the solution) | e.g., position, concentration | Real numbers |
Separable ODE: dy/dx = f(x)g(y)
A separable differential equation can be written in the form $ \frac{dy}{dx} = f(x)g(y) $. The core idea is to separate the variables, placing all terms involving y on one side and all terms involving x on the other: $ \frac{dy}{g(y)} = f(x)dx $. Once separated, we integrate both sides: $ \int \frac{1}{g(y)} dy = \int f(x) dx $. This results in an equation relating x and y, often implicitly. If possible, we solve for y to get the explicit solution: $ y(x) $. A constant of integration, C, is added during the integration process.
Variables:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| f(x) | Function of the independent variable x | Depends on context | Real numbers |
| g(y) | Function of the dependent variable y | Depends on context | Real numbers |
| C | Constant of Integration | N/A | Real numbers |
| x | Independent variable | e.g., time, position | Real numbers |
| y(x) | Dependent variable (the solution) | e.g., population, velocity | Real numbers |
Second-Order Linear ODE with Constant Coefficients: ay" + by' + cy = 0
For homogeneous second-order linear ODEs with constant coefficients, $ ay" + by' + cy = 0 $, we assume a solution of the form $ y = e^{rx} $. Substituting this into the equation yields the characteristic (or auxiliary) equation: $ ar^2 + br + c = 0 $. The roots of this quadratic equation, r1 and r2, determine the form of the general solution:
- Distinct Real Roots (r1 ≠ r2): $ y(x) = C_1e^{r_1x} + C_2e^{r_2x} $
- Repeated Real Roots (r1 = r2): $ y(x) = C_1e^{r_1x} + C_2xe^{r_1x} $
- Complex Conjugate Roots (r = α ± iβ): $ y(x) = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x)) $
Here, $ C_1 $ and $ C_2 $ are constants of integration determined by initial or boundary conditions.
Variables:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a, b, c | Constant coefficients | Depends on context | Real numbers (a ≠ 0) |
| r1, r2 | Roots of the characteristic equation | N/A | Real or Complex numbers |
| α, β | Real and imaginary parts of complex roots | N/A | Real numbers |
| C1, C2 | Constants of Integration | N/A | Real numbers |
| x | Independent variable | e.g., time, position | Real numbers |
| y(x) | Dependent variable (the solution) | e.g., displacement, temperature | Real numbers |
Practical Examples (Real-World Use Cases)
Example 1: Population Growth (Separable ODE)
A common application of separable differential equations is modeling population growth. Assume a population P grows at a rate proportional to its current size. This can be modeled by $ \frac{dP}{dt} = kP $, where P is the population size, t is time, and k is the growth rate constant.
Inputs:
- Initial Population ($ P_0 $): 1000
- Growth Rate (k): 0.05 per year
- Time (t): 10 years
Calculation:
This is a separable equation. $ \frac{dP}{P} = k dt $. Integrating both sides: $ \int \frac{dP}{P} = \int k dt \implies \ln|P| = kt + C $. Exponentiating: $ P = e^{kt+C} = e^C e^{kt} $. Let $ A = e^C $. So, $ P(t) = Ae^{kt} $. Using the initial condition $ P(0) = 1000 $: $ 1000 = Ae^{k \cdot 0} \implies A = 1000 $. The specific solution is $ P(t) = 1000e^{0.05t} $.
At t = 10 years: $ P(10) = 1000e^{0.05 \times 10} = 1000e^{0.5} \approx 1000 \times 1.6487 = 1648.7 $.
Result: The population is predicted to be approximately 1649 individuals after 10 years.
Interpretation: This model shows exponential growth, where the population increases at an accelerating rate due to the constant relative growth rate.
Example 2: Simple Harmonic Motion (Second-Order Linear ODE)
Consider a mass m attached to a spring with spring constant k, undergoing simple harmonic motion without damping. The equation of motion is $ m\frac{d^2x}{dt^2} + kx = 0 $, or $ \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 $. Let $ \omega^2 = \frac{k}{m} $, so the equation is $ x" + \omega^2 x = 0 $.
Inputs:
- Coefficient a: 1
- Coefficient b: 0
- Coefficient c: 4 ($ \omega^2 = 4 $)
- Initial Position ($ x(0) $): 2
- Initial Velocity ($ x'(0) $): 0
Calculation:
The characteristic equation is $ r^2 + 4 = 0 $. The roots are $ r = \pm\sqrt{-4} = \pm 2i $. This is the complex conjugate case with $ \alpha = 0 $ and $ \beta = 2 $. The general solution is $ x(t) = e^{0t}(C_1\cos(2t) + C_2\sin(2t)) = C_1\cos(2t) + C_2\sin(2t) $.
Using initial conditions:
- $ x(0) = 2 \implies C_1\cos(0) + C_2\sin(0) = 2 \implies C_1 = 2 $.
- The velocity is $ x'(t) = -2C_1\sin(2t) + 2C_2\cos(2t) $.
- $ x'(0) = 0 \implies -2C_1\sin(0) + 2C_2\cos(0) = 0 \implies 2C_2 = 0 \implies C_2 = 0 $.
The specific solution is $ x(t) = 2\cos(2t) $.
Result: The position of the mass at time t is given by $ x(t) = 2\cos(2t) $. This represents simple harmonic motion with amplitude 2 and angular frequency 2 rad/s.
Interpretation: The mass oscillates symmetrically around the equilibrium position (x=0) with a constant period, demonstrating the undamped oscillatory behavior characteristic of this type of differential equation.
How to Use This Differential Equation Calculator
Using this calculator is straightforward. Follow these steps to solve and visualize your differential equations:
- Select Equation Type: Choose the type of differential equation you wish to solve from the dropdown menu (e.g., Linear First-Order, Separable, Second-Order Linear with Constant Coefficients).
- Input Parameters: Based on the selected type, relevant input fields will appear. Enter the necessary coefficients, functions (as simple expressions if supported, or use numerical approximations), and initial/boundary conditions. For example, for $ ay" + by' + cy = 0 $, you'll input values for a, b, and c. For $ \frac{dP}{dt} = kP $, you'd input k and an initial P.
- Calculate: Click the "Calculate" button. The calculator will process your inputs using the appropriate mathematical methods.
- Interpret Results:
- Primary Result: The main solution (e.g., the explicit function y(x) or a key value like population at time t) will be prominently displayed.
- Intermediate Values: Key components of the solution, such as the integrating factor, roots of the characteristic equation, or constants of integration, will be listed.
- Formula Explanation: A brief description of the mathematical formula or method used for the selected equation type is provided.
- Table: A table summarizes the input parameters, calculated values, and their units.
- Chart: A dynamic chart visualizes the solution, showing how the dependent variable changes with respect to the independent variable.
- Copy Results: Use the "Copy Results" button to easily transfer the main result, intermediate values, and key assumptions to your clipboard for use in reports or further analysis.
- Reset: Click "Reset" to clear all fields and return to default settings, allowing you to start a new calculation.
Decision-Making Guidance: The results provide insights into the behavior of the system modeled by the differential equation. For instance, population growth models help predict future sizes, while motion equations describe an object's trajectory. Compare different scenarios by adjusting input parameters to understand how changes affect the outcome.
Key Factors That Affect Differential Equation Results
Several factors significantly influence the solution and interpretation of differential equations:
- Initial Conditions / Boundary Conditions: These are the specific values of the function and/or its derivatives at a particular point (or points). They are crucial for determining the unique constants of integration ($C_1, C_2$, etc.), leading to a specific solution from a family of possible solutions. For example, the initial position and velocity of a mass on a spring dictate its exact motion.
- Coefficients and Parameters: The constants within the differential equation (like a, b, c in $ay"+by'+c=0$, or k in population growth) directly shape the solution's behavior. Changing these parameters can drastically alter the outcome, such as changing oscillation frequency, growth rate, or decay speed.
- Type of Equation: The structure of the differential equation (linear vs. nonlinear, order, homogeneity) dictates the mathematical methods applicable and the nature of the solution. A linear equation might have predictable behavior, while a nonlinear one could exhibit chaotic or complex dynamics.
- Domain of the Independent Variable: The range of values for the independent variable (often time or position) over which the solution is considered is important. Solutions might be valid only for positive time, or within a specific spatial range. Extrapolating beyond the valid domain can lead to incorrect conclusions.
- Assumptions of the Model: Real-world phenomena are often simplified to fit differential equation models. Assumptions like constant growth rates, absence of friction/damping, or uniform distribution might not hold true indefinitely. Recognizing these assumptions is key to understanding the limitations of the model's predictions.
- Numerical Stability (for numerical solutions): When analytical solutions are intractable, numerical methods are used. The choice of method, step size, and algorithm can affect the accuracy and stability of the computed solution. Small errors in calculation can sometimes amplify over time.
- External Forcing Functions: In non-homogeneous equations ($ay"+by'+c=f(x)$), the term $f(x)$ represents external influences (like a driving force or a changing environmental factor). The nature and magnitude of this forcing function directly impact the system's response.
- Discretization Errors (in numerical methods): When continuous differential equations are approximated by discrete steps for computation, inherent errors arise. The magnitude of these errors depends on the step size and the complexity of the solution curve.
Frequently Asked Questions (FAQ)
What is the difference between an ordinary differential equation (ODE) and a partial differential equation (PDE)?
An ODE involves derivatives of a function with respect to only one independent variable, like $dy/dx$. A PDE involves partial derivatives with respect to two or more independent variables, such as $ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} $. This calculator focuses on ODEs.
Can this calculator solve any differential equation?
No, this calculator is designed for specific, common types of ordinary differential equations (linear first-order, separable, and second-order linear with constant coefficients). Many other types exist, including nonlinear and higher-order equations, which require different solution techniques.
What does the "constant of integration" represent?
The constant of integration (C) arises when integrating. It represents a family of possible solutions. Initial or boundary conditions are needed to determine a specific value for C and find a unique solution.
How are differential equations used in finance?
Differential equations model financial phenomena like option pricing (e.g., Black-Scholes equation), interest rate dynamics, and portfolio optimization. They help understand how financial variables change over time.
What is the characteristic equation for a second-order ODE?
For an equation of the form $ ay" + by' + cy = 0 $, the characteristic equation is $ ar^2 + br + c = 0 $. Solving this quadratic equation for 'r' provides the basis for finding the general solution.
Can I input complex functions for P(x) or Q(x)?
This calculator is primarily designed for simpler functional forms or numerical coefficients. For highly complex or symbolic functions, specialized symbolic math software is typically required.
What does the chart show?
The chart visualizes the solution y(x) (or P(t), x(t), etc.) against the independent variable (x or t). It helps to understand the behavior, trends, and oscillations of the system being modeled.
How do I interpret the "intermediate values"?
Intermediate values are key components derived during the solution process, such as the integrating factor, roots of the characteristic equation, or constants of integration. They provide insight into the structure of the solution and the underlying mathematical steps.
Related Tools and Internal Resources
- Linear First-Order ODE Solver Understand and solve equations of the form dy/dx + P(x)y = Q(x).
- Separable ODE Calculator Solve differential equations where variables can be separated.
- Second-Order ODE Solver Analyze systems described by ay" + by' + cy = 0.
- Numerical Methods for ODEs Explore advanced techniques for approximating solutions.
- Calculus Fundamentals Review essential concepts like derivatives and integrals.
- Physics Modeling Tools Discover calculators for physical systems governed by differential equations.
Enter the constant coefficient for P(x). For simplicity, assume P(x) is a constant.
Enter the constant value for Q(x). For simplicity, assume Q(x) is a constant.
The starting value for the independent variable x.
The value of y at the initial x.
The upper limit for the independent variable x for visualization.
`;
populateTableLinearFirstOrder();
} else if (type === 'ode_separable') {
formulaTextElement.innerHTML = 'Formula Used: $ \\frac{dy}{dx} = f(x)g(y) $. Solution involves separating variables: $ \\frac{dy}{g(y)} = f(x)dx $. Integrate both sides: $ \\int \\frac{1}{g(y)} dy = \\int f(x) dx $.';
inputSection.innerHTML = `
Enter the constant value for f(x). For simplicity, assume f(x) is a constant.
Enter the constant coefficient for g(y). For simplicity, assume g(y) = ky.
The starting value for the independent variable x.
The value of y at the initial x.
The upper limit for the independent variable x for visualization.
`;
populateTableSeparable();
} else if (type === 'ode_second_order_linear_constant_coeff') {
formulaTextElement.innerHTML = 'Formula Used: $ ay\'\' + by\' + cy = 0 $. Solved via characteristic equation $ ar^2 + br + c = 0 $. Solution depends on roots $ r_1, r_2 $.';
inputSection.innerHTML = `
Coefficient of the second derivative (y"). Must not be zero.
Coefficient of the first derivative (y').
Coefficient of the dependent variable (y).
The starting value for the independent variable x.
The value of y at the initial x.
The value of the first derivative of y at the initial x.
The upper limit for the independent variable x for visualization.
`;
populateTableSecondOrder();
}
// Clear previous results and chart
document.getElementById('highlightedResult').innerHTML = 'Enter values to see results.';
document.getElementById('intermediateResults').innerHTML = ";
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
}
function validateInput(id, min = null, max = null) {
var input = document.getElementById(id);
var value = input.value.trim();
var errorElement = document.getElementById(id + '_error');
errorElement.style.display = 'none'; // Hide error by default
if (value === ") {
errorElement.textContent = 'This field cannot be empty.';
errorElement.style.display = 'block';
return false;
}
var numValue = parseFloat(value);
if (isNaN(numValue)) {
errorElement.textContent = 'Please enter a valid number.';
errorElement.style.display = 'block';
return false;
}
if (min !== null && numValue max) {
errorElement.textContent = 'Value must be no more than ' + max + '.';
errorElement.style.display = 'block';
return false;
}
return true;
}
function calculateDifferentialEquation() {
var isValid = true;
var resultsDiv = document.getElementById('results');
var highlightedResultDiv = document.getElementById('highlightedResult');
var intermediateResultsDiv = document.getElementById('intermediateResults');
var chartContainer = document.getElementById('chartContainer');
var canvas = document.getElementById('differentialEquationChart');
var ctx = canvas.getContext('2d');
// Clear previous chart
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
chartContainer.style.display = 'none'; // Hide chart until new data is available
// Reset error messages
var errorElements = document.querySelectorAll('.error-message');
for (var i = 0; i < errorElements.length; i++) {
errorElements[i].style.display = 'none';
}
if (currentEquationType === 'ode_linear_first_order') {
isValid &= validateInput('p_x_coeff');
isValid &= validateInput('q_x_func');
isValid &= validateInput('initial_x');
isValid &= validateInput('initial_y');
isValid &= validateInput('calc_x_end');
if (!isValid) return;
var p_x = parseFloat(document.getElementById('p_x_coeff').value);
var q_x = parseFloat(document.getElementById('q_x_func').value);
var x0 = parseFloat(document.getElementById('initial_x').value);
var y0 = parseFloat(document.getElementById('initial_y').value);
var x_end = parseFloat(document.getElementById('calc_x_end').value);
if (p_x === 0) { // Simple case: dy/dx = Q(x)
var integral_qx = q_x * (x_end – x0);
var y_final = y0 + integral_qx;
highlightedResultDiv.innerHTML = 'y(' + x_end + ') = ' + y_final.toFixed(4);
intermediateResultsDiv.innerHTML = `
Integrating Factor (μ(x)): N/A (P(x)=0)
Integral of μ(x)Q(x) dx: ${integral_qx.toFixed(4)}
Constant of Integration (C): ${y0.toFixed(4)}
`;
updateTableLinearFirstOrder(p_x, q_x, x0, y0, y_final);
plotLinearFirstOrder(p_x, q_x, x0, y0, x_end);
} else {
// Calculate integrating factor: mu = exp(integral(P(x)dx)) = exp(P*x) for constant P
var mu_x = Math.exp(p_x * x0); // Evaluate at x0 for the formula structure, though it's constant
// Integral of mu(x)*Q(x) dx = integral(exp(p*x)*q_x dx) = (q_x / p_x) * exp(p*x)
var integral_mu_qx_at_x0 = (q_x / p_x) * Math.exp(p_x * x0);
var integral_mu_qx_at_x_end = (q_x / p_x) * Math.exp(p_x * x_end);
var integral_diff = integral_mu_qx_at_x_end – integral_mu_qx_at_x0;
// y(x) = (1/mu(x)) * (integral(mu(x)Q(x)dx) + C)
// y(x0) = (1/mu(x0)) * (integral_mu_qx_at_x0 + C) => C = y0*mu(x0) – integral_mu_qx_at_x0
var C = y0 * mu_x – integral_mu_qx_at_x0;
// y(x_end) = (1/mu(x_end)) * (integral_mu_qx_at_x_end + C)
var mu_x_end = Math.exp(p_x * x_end);
var y_final = (1 / mu_x_end) * (integral_mu_qx_at_x_end + C);
highlightedResultDiv.innerHTML = 'y(' + x_end + ') = ' + y_final.toFixed(4);
intermediateResultsDiv.innerHTML = `
Integrating Factor (μ(x)): ${mu_x.toFixed(4)} (evaluated at x=${x0})
Integral of μ(x)Q(x) dx from ${x0} to ${x_end}: ${integral_diff.toFixed(4)}
Constant of Integration (C): ${C.toFixed(4)}
`;
updateTableLinearFirstOrder(p_x, q_x, x0, y0, y_final);
plotLinearFirstOrder(p_x, q_x, x0, y0, x_end);
}
resultsDiv.style.display = 'block';
} else if (currentEquationType === 'ode_separable') {
isValid &= validateInput('f_x_func');
isValid &= validateInput('g_y_func');
isValid &= validateInput('initial_x_sep');
isValid &= validateInput('initial_y_sep');
isValid &= validateInput('calc_x_end_sep');
if (!isValid) return;
var f_x_val = parseFloat(document.getElementById('f_x_func').value); // Assuming constant f(x)
var g_y_coeff = parseFloat(document.getElementById('g_y_func').value); // Assuming g(y) = ky
var x0_sep = parseFloat(document.getElementById('initial_x_sep').value);
var y0_sep = parseFloat(document.getElementById('initial_y_sep').value);
var x_end_sep = parseFloat(document.getElementById('calc_x_end_sep').value);
// dy/dx = f_x_val * g_y_coeff * y
// dy/y = (f_x_val * g_y_coeff) dx
// ln|y| = (f_x_val * g_y_coeff) * x + C
// y(x) = A * exp((f_x_val * g_y_coeff) * x)
var k_sep = f_x_val * g_y_coeff; // Combined constant rate
// Find A using initial condition y0 = A * exp(k_sep * x0)
var A = y0_sep / Math.exp(k_sep * x0_sep);
var y_final_sep = A * Math.exp(k_sep * x_end_sep);
highlightedResultDiv.innerHTML = 'y(' + x_end_sep + ') = ' + y_final_sep.toFixed(4);
intermediateResultsDiv.innerHTML = `
Combined Rate (k = f(x)*g'(y)): ${k_sep.toFixed(4)}
Constant Multiplier (A): ${A.toFixed(4)}
Solution Form: y(x) = A * exp(kx)
`;
updateTableSeparable(f_x_val, g_y_coeff, x0_sep, y0_sep, y_final_sep);
plotSeparable(k_sep, A, x0_sep, x_end_sep);
resultsDiv.style.display = 'block';
} else if (currentEquationType === 'ode_second_order_linear_constant_coeff') {
isValid &= validateInput('coeff_a', null, null); // a cannot be 0
if (parseFloat(document.getElementById('coeff_a').value) === 0) {
document.getElementById('coeff_a_error').textContent = 'Coefficient a cannot be zero for a second-order ODE.';
document.getElementById('coeff_a_error').style.display = 'block';
isValid = false;
}
isValid &= validateInput('coeff_b');
isValid &= validateInput('coeff_c');
isValid &= validateInput('initial_x_2nd');
isValid &= validateInput('initial_y_2nd');
isValid &= validateInput('initial_y_prime_2nd');
isValid &= validateInput('calc_x_end_2nd');
if (!isValid) return;
var a = parseFloat(document.getElementById('coeff_a').value);
var b = parseFloat(document.getElementById('coeff_b').value);
var c = parseFloat(document.getElementById('coeff_c').value);
var x0_2nd = parseFloat(document.getElementById('initial_x_2nd').value);
var y0_2nd = parseFloat(document.getElementById('initial_y_2nd').value);
var y_prime0_2nd = parseFloat(document.getElementById('initial_y_prime_2nd').value);
var x_end_2nd = parseFloat(document.getElementById('calc_x_end_2nd').value);
// Solve characteristic equation: ar^2 + br + c = 0
var discriminant = b * b – 4 * a * c;
var r1, r2, alpha, beta;
var solutionType = "";
var generalSolutionText = "";
if (discriminant > 0) { // Distinct real roots
r1 = (-b + Math.sqrt(discriminant)) / (2 * a);
r2 = (-b – Math.sqrt(discriminant)) / (2 * a);
solutionType = "Distinct Real Roots";
generalSolutionText = `y(x) = C1*exp(${r1.toFixed(4)}*x) + C2*exp(${r2.toFixed(4)}*x)`;
} else if (discriminant === 0) { // Repeated real roots
r1 = -b / (2 * a);
r2 = r1; // Same root
solutionType = "Repeated Real Roots";
generalSolutionText = `y(x) = C1*exp(${r1.toFixed(4)}*x) + C2*x*exp(${r1.toFixed(4)}*x)`;
} else { // Complex conjugate roots
alpha = -b / (2 * a);
beta = Math.sqrt(-discriminant) / (2 * a);
solutionType = "Complex Conjugate Roots";
generalSolutionText = `y(x) = exp(${alpha.toFixed(4)}*x) * (C1*cos(${beta.toFixed(4)}*x) + C2*sin(${beta.toFixed(4)}*x))`;
}
var C1, C2;
var y_final_2nd;
// Use initial conditions to find C1 and C2
if (discriminant >= 0) { // Real roots (distinct or repeated)
var exp_r1_x0 = Math.exp(r1 * x0_2nd);
var exp_r2_x0 = Math.exp(r2 * x0_2nd);
var x_exp_r1_x0 = (discriminant === 0) ? x0_2nd * exp_r1_x0 : 0; // Only for repeated roots
var exp_r1_x_prime = r1 * exp_r1_x0;
var exp_r2_x_prime = r2 * exp_r2_x0;
var x_exp_r1_x_prime_plus_exp_r1_x0 = (discriminant === 0) ? (exp_r1_x0 + r1 * x_exp_r1_x0) : 0; // Derivative term for repeated roots
if (discriminant > 0) { // Distinct real roots
// y0 = C1*exp(r1*x0) + C2*exp(r2*x0)
// y'0 = C1*r1*exp(r1*x0) + C2*r2*exp(r2*x0)
var det = exp_r1_x0 * r2 * exp_r2_x0 – exp_r2_x0 * r1 * exp_r1_x0;
C1 = (y0_2nd * r2 * exp_r2_x0 – y_prime0_2nd * exp_r2_x0) / det;
C2 = (y_prime0_2nd * exp_r1_x0 – y0_2nd * r1 * exp_r1_x0) / det;
} else { // Repeated real roots (r1 = r2 = r)
// y0 = C1*exp(r*x0) + C2*x0*exp(r*x0)
// y'0 = C1*r*exp(r*x0) + C2*(exp(r*x0) + r*x0*exp(r*x0))
var exp_r_x0 = Math.exp(r1 * x0_2nd);
var denom = exp_r_x0 * exp_r_x0; // Simplified denominator for C1, C2 calculation
C1 = (y0_2nd * (exp_r_x0 + r1 * x0_2nd * exp_r_x0) – y_prime0_2nd * x0_2nd * exp_r_x0) / denom;
C2 = (y_prime0_2nd * exp_r_x0 – y0_2nd * r1 * exp_r_x0) / denom;
}
} else { // Complex conjugate roots (r = alpha +/- i*beta)
alpha = -b / (2 * a);
beta = Math.sqrt(-discriminant) / (2 * a);
var exp_alpha_x0 = Math.exp(alpha * x0_2nd);
var cos_beta_x0 = Math.cos(beta * x0_2nd);
var sin_beta_x0 = Math.sin(beta * x0_2nd);
// y0 = exp(alpha*x0) * (C1*cos(beta*x0) + C2*sin(beta*x0))
// y'0 = alpha*exp(alpha*x0)*(C1*cos(beta*x0) + C2*sin(beta*x0)) + exp(alpha*x0)*(-C1*beta*sin(beta*x0) + C2*beta*cos(beta*x0))
// y'0 = alpha*y0 + exp(alpha*x0)*(-C1*beta*sin(beta*x0) + C2*beta*cos(beta*x0))
var term1_y0 = exp_alpha_x0 * cos_beta_x0;
var term2_y0 = exp_alpha_x0 * sin_beta_x0;
var term1_y_prime0 = alpha * exp_alpha_x0 * cos_beta_x0 – beta * exp_alpha_x0 * sin_beta_x0;
var term2_y_prime0 = alpha * exp_alpha_x0 * sin_beta_x0 + beta * exp_alpha_x0 * cos_beta_x0;
// Solve system:
// y0 = C1 * term1_y0 + C2 * term2_y0
// y'0 = C1 * term1_y_prime0 + C2 * term2_y_prime0
var det = term1_y0 * term2_y_prime0 – term2_y0 * term1_y_prime0;
C1 = (y0_2nd * term2_y_prime0 – y_prime0_2nd * term2_y0) / det;
C2 = (y_prime0_2nd * term1_y0 – y0_2nd * term1_y_prime0) / det;
}
// Calculate final value y(x_end)
if (discriminant >= 0) {
var exp_r1_x_end = Math.exp(r1 * x_end_2nd);
var exp_r2_x_end = (discriminant === 0) ? x_end_2nd * exp_r1_x_end : Math.exp(r2 * x_end_2nd);
y_final_2nd = C1 * exp_r1_x_end + C2 * exp_r2_x_end;
} else {
var exp_alpha_x_end = Math.exp(alpha * x_end_2nd);
var cos_beta_x_end = Math.cos(beta * x_end_2nd);
var sin_beta_x_end = Math.sin(beta * x_end_2nd);
y_final_2nd = exp_alpha_x_end * (C1 * cos_beta_x_end + C2 * sin_beta_x_end);
}
highlightedResultDiv.innerHTML = 'y(' + x_end_2nd + ') = ' + y_final_2nd.toFixed(4);
intermediateResultsDiv.innerHTML = `
Characteristic Equation Roots: ${solutionType} (r1=${r1 ? r1.toFixed(4) : "}, r2=${r2 ? r2.toFixed(4) : "}, α=${alpha !== undefined ? alpha.toFixed(4) : "}, β=${beta !== undefined ? beta.toFixed(4) : "})
General Solution Form: ${generalSolutionText}
Constant C1: ${C1.toFixed(4)}
Constant C2: ${C2.toFixed(4)}
`;
updateTableSecondOrder(a, b, c, x0_2nd, y0_2nd, y_prime0_2nd, y_final_2nd);
plotSecondOrder(a, b, c, x0_2nd, y0_2nd, y_prime0_2nd, x_end_2nd);
resultsDiv.style.display = 'block';
}
}
function resetForm() {
document.getElementById('equationType').value = 'ode_linear_first_order';
updateInputs(); // This will reset all fields to their defaults
// Ensure results are hidden until calculation
document.getElementById('results').style.display = 'none';
}
function copyResults() {
var highlightedResult = document.getElementById('highlightedResult').innerText;
var intermediateResults = document.getElementById('intermediateResults').innerText.replace('Intermediate Results', ");
var formulaText = document.getElementById('formulaText').innerText.replace('Formula Used:', ");
var tableHtml = document.getElementById('resultsTable').outerHTML;
var contentToCopy = `— Differential Equation Calculation Results —\n\n`;
contentToCopy += `Primary Result:\n${highlightedResult}\n\n`;
contentToCopy += `Intermediate Values:\n${intermediateResults}\n\n`;
contentToCopy += `Formula Explanation:\n${formulaText}\n\n`;
contentToCopy += `Key Parameters:\n${tableHtml.replace(/]*>/g, ").replace(/ /g, ' ')}\n\n`; // Basic text extraction from table
contentToCopy += `— End of Results —`;
// Use a temporary textarea to copy to clipboard
var tempTextArea = document.createElement("textarea");
tempTextArea.value = contentToCopy;
tempTextArea.style.position = "fixed"; // Avoid scrolling to bottom of page in MS Edge.
tempTextArea.style.left = "-9999px";
tempTextArea.style.top = "-9999px";
document.body.appendChild(tempTextArea);
tempTextArea.focus();
tempTextArea.select();
try {
var successful = document.execCommand('copy');
var msg = successful ? 'Results copied to clipboard!' : 'Copying failed!';
// Optionally show a temporary message to the user
var copyButton = document.querySelector('button.success');
var originalText = copyButton.innerText;
copyButton.innerText = msg;
setTimeout(function() {
copyButton.innerText = originalText;
}, 2000);
} catch (err) {
console.error('Fallback: Oops, unable to copy', err);
var copyButton = document.querySelector('button.success');
var originalText = copyButton.innerText;
copyButton.innerText = 'Copy Failed!';
setTimeout(function() {
copyButton.innerText = originalText;
}, 2000);
}
document.body.removeChild(tempTextArea);
}
// — Table Population Functions —
function populateTableLinearFirstOrder() {
var tbody = document.getElementById('tableBody');
tbody.innerHTML = `