Separation of Variables Calculator

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Solve Ordinary Differential Equations with Ease

Separation of Variables Calculator

Calculation Results

Formula Used: For y’ = f(x)g(y), we separate variables to get dy/g(y) = f(x)dx. Integrating both sides gives ∫(1/g(y)) dy = ∫f(x) dx + C. The solution y(x) is found by solving this implicit equation for y. For M(x)dx + N(y)dy = 0, it’s already separated.

What is Separation of Variables?

Separation of variables is a fundamental mathematical technique used primarily to solve ordinary differential equations (ODEs). It’s a powerful method that simplifies complex differential equations by rearranging them into a form where terms involving the dependent variable (often ‘y’) can be isolated on one side of the equation, and terms involving the independent variable (often ‘x’) can be isolated on the other. This allows us to integrate both sides independently, leading to a solution for the differential equation.

Who should use it: This technique is essential for students and professionals in fields like physics, engineering, economics, biology, and mathematics who encounter differential equations in their work. It’s a cornerstone of calculus II and differential equations courses.

Common misconceptions: A frequent misunderstanding is that all ODEs can be solved using separation of variables. This is not true; the technique only applies to ODEs that can be algebraically manipulated into the specific form y’ = f(x)g(y) or M(x)dx + N(y)dy = 0. Another misconception is that the integration constants are always ignored; they are crucial for finding the particular solution that fits initial conditions.

Separation of Variables Formula and Mathematical Explanation

The core idea behind the separation of variables method is to transform a differential equation into an integrable form. Let’s consider the two common forms:

Form 1: y’ = f(x)g(y)

This is the most direct form where the derivative of y with respect to x is a product of a function of x and a function of y.

1. Rewrite the derivative: Replace y’ with dy/dx.
   
   dy/dx = f(x)g(y)
2. Separate the variables: Multiply both sides by dx and divide by g(y) (assuming g(y) ≠ 0).
   
   (1/g(y)) dy = f(x) dx
3. Integrate both sides: Integrate the left side with respect to y and the right side with respect to x.
   
   ∫(1/g(y)) dy = ∫f(x) dx
4. Add the constant of integration: Remember to add a constant of integration, C, to one side (usually the side with the independent variable).
   
   ∫(1/g(y)) dy = ∫f(x) dx + C
5. Solve for y (if possible): Evaluate the integrals. Let F(x) be the antiderivative of f(x) and G(y) be the antiderivative of 1/g(y). The equation becomes:
   
   G(y) = F(x) + C
   Then, solve this implicit equation for y in terms of x to find the general solution. If initial conditions (e.g., y(x₀) = y₀) are provided, substitute them into the general solution to find the specific value of C, yielding the particular solution.

Form 2: M(x)dx + N(y)dy = 0

This form is already partially separated. If M is a function of x only and N is a function of y only, the equation is directly integrable.

1. Check if separable: Verify that M depends only on x and N depends only on y.
2. Integrate both sides:
   
   ∫M(x) dx + ∫N(y) dy = C
3. Solve for y (if possible): Evaluate the integrals and solve for y if required.

If M depends on both x and y, and N depends on both x and y, the equation might still be separable if it can be rewritten as f(x)dx + g(y)dy = 0. For example, if M(x,y) = M₁(x)M₂(y) and N(x,y) = N₁(x)N₂(y), we can rearrange it.

Variables Table

Key Variables and Their Meanings

Variable	Meaning	Unit	Typical Range
y	Dependent Variable	Depends on context (e.g., position, concentration)	Real numbers
x	Independent Variable	Depends on context (e.g., time, distance)	Real numbers
y’ or dy/dx	First Derivative of y with respect to x	Units of y / Units of x	Real numbers
f(x)	Function of the independent variable	Depends on context	Real numbers
g(y)	Function of the dependent variable	Depends on context	Real numbers
M(x)	Function of x in M(x)dx	Depends on context	Real numbers
N(y)	Function of y in N(y)dy	Depends on context	Real numbers
C	Constant of Integration	Depends on context	Real numbers

Practical Examples (Real-World Use Cases)

Example 1: Population Growth

Consider a population whose growth rate is proportional to its current size. This can be modeled by the ODE: dy/dt = ky, where y is the population size and t is time.

Inputs:

• Equation Type: y' = f(x)g(y)
• f(x): k (a constant, e.g., 0.02)
• g(y): y
• Initial x Value (t₀): 0
• Initial y Value (y₀): 1000 (initial population)
• Evaluate at x (t): 10 (years)

Calculation Steps (Conceptual):

1. Separate: (1/y) dy = k dt
2. Integrate: ∫(1/y) dy = ∫k dt
3. Result: ln|y| = kt + C
4. Solve for y: y = Ae^(kt) (where A = e^C)
5. Apply initial condition y(0) = 1000: 1000 = Ae^(k*0) => A = 1000
6. Particular Solution: y(t) = 1000 * e^(kt)
7. Evaluate at t=10: y(10) = 1000 * e^(0.02 * 10) = 1000 * e^0.2 ≈ 1221.4

Calculator Output Interpretation: The calculator would find the value of y at t=10, approximately 1221. This means that with an initial population of 1000 and a growth rate constant k=0.02, the population is predicted to reach about 1221 after 10 years.

Example 2: Radioactive Decay

The rate of decay of a radioactive substance is proportional to the amount of substance present. The ODE is: dA/dt = -λA, where A is the amount of substance and λ is the decay constant.

Inputs:

• Equation Type: y' = f(x)g(y)
• f(x): -λ (a constant, e.g., -0.05)
• g(y): A
• Initial x Value (t₀): 0
• Initial y Value (y₀): 500 (initial amount in grams)
• Evaluate at x (t): 20 (hours)

Calculation Steps (Conceptual):

1. Separate: (1/A) dA = -λ dt
2. Integrate: ∫(1/A) dA = ∫-λ dt
3. Result: ln|A| = -λt + C
4. Solve for A: A = Be^(-λt) (where B = e^C)
5. Apply initial condition A(0) = 500: 500 = Be^(-λ*0) => B = 500
6. Particular Solution: A(t) = 500 * e^(-λt)
7. Evaluate at t=20: A(20) = 500 * e^(-0.05 * 20) = 500 * e^-1 ≈ 183.9

Calculator Output Interpretation: The calculator would estimate that after 20 hours, approximately 183.9 grams of the substance remain. This is crucial for understanding half-life and decay rates in nuclear physics and related fields.

How to Use This Separation of Variables Calculator

Our Separation of Variables Calculator is designed to be intuitive and provide quick solutions to common ODEs. Follow these steps:

1. Select Equation Type: Choose the form that best matches your differential equation: y' = f(x)g(y) or M(x)dx + N(y)dy = 0.
2. Input Functions:
   • For y' = f(x)g(y): Enter the expressions for f(x) and g(y). Use standard mathematical notation (e.g., x^2 for x squared, exp(x) for e^x, sin(x), cos(x), y^3).
   • For M(x)dx + N(y)dy = 0: Enter the expressions for M(x) and N(y).
3. Enter Initial Conditions: Input the starting values for x (x₀) and y (y₀). These are essential for finding the particular solution.
4. Specify Evaluation Point: Enter the value of x at which you want to find the corresponding y value.
5. Calculate: Click the “Calculate Solution” button.

Reading the Results:

• Primary Result: This is the calculated value of y at your specified x-value.
• Intermediate Values: These show the results of integrating f(x) and 1/g(y) (or M(x) and N(y)), and the calculated constant of integration C based on your initial conditions.
• Formula Explanation: Provides a brief overview of the mathematical steps involved.
• Solution Curve: The chart visualizes the solution, showing the relationship between x and y, including your initial point and the calculated solution point.

Decision-Making Guidance: Use the calculated y-value to predict future states, analyze system behavior, or verify theoretical models. For instance, in population dynamics, it predicts future population size; in physics, it might predict the position of an object over time.

Key Factors That Affect Separation of Variables Results

While the separation of variables method is powerful, several factors influence the accuracy and applicability of the results:

1. Correctness of the ODE Model: The differential equation itself must accurately represent the physical, biological, or financial system being studied. An inaccurate model will lead to inaccurate predictions, regardless of the solution method.
2. Form of the Functions f(x), g(y), M(x), N(y): The technique is only applicable if the ODE can be algebraically rearranged into the separable form. If the functions are too complex or intertwined, separation might not be possible.
3. Integrability of Functions: Even if separable, the integrals ∫f(x) dx and ∫(1/g(y)) dy (or ∫M(x) dx and ∫N(y) dy) must be solvable. Some functions do not have elementary antiderivatives, requiring numerical methods or advanced integration techniques.
4. Initial Conditions (y₀, x₀): These are critical for determining the specific solution. Different initial conditions lead to different particular solutions, even for the same differential equation. The calculator uses these to find the constant C.
5. Domain of Validity: Solutions derived using separation of variables might only be valid over a specific range of x and y. For example, dividing by g(y) assumes g(y) ≠ 0. If g(y) = 0 for some y, those constant values might also be solutions (equilibrium solutions).
6. Numerical Precision: When dealing with complex functions or large numbers, the precision of the calculation can affect the final result. Our calculator uses standard floating-point arithmetic.
7. Assumptions of the Model: Many models assume ideal conditions (e.g., no friction, constant rates, no external influences). Real-world scenarios often involve complexities not captured by the basic ODE, impacting the practical relevance of the calculated results.
8. Units Consistency: Ensure that the units used for x, y, and constants are consistent throughout the problem. Mismatched units can lead to nonsensical results.

Frequently Asked Questions (FAQ)

• Q1: Can all differential equations be solved using separation of variables?
  A: No. Only ODEs that can be algebraically rearranged into the form y’ = f(x)g(y) or M(x)dx + N(y)dy = 0 are solvable by this method.
• Q2: What if g(y) = 0?
  A: If g(y) = 0 for some value(s) of y, those constant values y = constant are often equilibrium solutions. You should check them separately.
• Q3: Do I need initial conditions?
  A: Initial conditions (like y(x₀) = y₀) are needed to find a *particular* solution. Without them, you get the *general* solution containing the arbitrary constant C.
• Q4: What if the integrals are too difficult to solve?
  A: If the integrals ∫f(x) dx or ∫(1/g(y)) dy cannot be solved analytically, numerical methods (like Euler’s method or Runge-Kutta) are required. This calculator handles analytically solvable integrals.
• Q5: How does the calculator handle complex functions like trigonometric or exponential?
  A: The calculator uses JavaScript’s built-in Math functions (e.g., Math.sin, Math.exp). Ensure your input uses standard notation like `sin(x)` or `exp(x)`.
• Q6: What does the chart represent?
  A: The chart plots the solution curve y(x) based on the calculated particular solution. It helps visualize the behavior of the system over the specified range.
• Q7: Can this method be used for partial differential equations (PDEs)?
  A: Yes, a similar technique called “separation of variables” is also used for solving certain PDEs, but it involves separating variables in multiple dimensions and is more complex. This calculator focuses on Ordinary Differential Equations (ODEs).
• Q8: What is the difference between the general and particular solution?
  A: The general solution includes an arbitrary constant (C) and represents a family of curves. The particular solution is a single curve from that family, determined by specific initial or boundary conditions.

Related Tools and Internal Resources

// — Global Variables —
var currentEquationType = ‘simple’;

// — Helper Functions —
function evaluateExpression(expression, xValue, yValue) {
try {
var scope = {};
if (typeof xValue !== ‘undefined’) scope.x = xValue;
if (typeof yValue !== ‘undefined’) scope.y = yValue;

// Replace common math functions with JavaScript equivalents
expression = expression.replace(/sin/g, ‘Math.sin’);
expression = expression.replace(/cos/g, ‘Math.cos’);
expression = expression.replace(/tan/g, ‘Math.tan’);
expression = expression.replace(/exp/g, ‘Math.exp’);
expression = expression.replace(/log/g, ‘Math.log’);
expression = expression.replace(/sqrt/g, ‘Math.sqrt’);
expression = expression.replace(/\^/g, ‘**’); // Exponentiation operator

// Basic safety check: disallow dangerous characters
if (/[^a-zA-Z0-9_()+\-*/.% ]/.test(expression)) {
throw new Error(“Invalid characters in expression.”);
}

// Use Function constructor for evaluation with scope
var func = new Function(‘Math’, ‘scope’, ‘with(scope) {‘ +
‘return ‘ + expression + ‘;’ +
‘}’);
var result = func(Math, scope);

if (isNaN(result) || !isFinite(result)) {
throw new Error(“Result is not a number.”);
}
return result;
} catch (e) {
console.error(“Evaluation error:”, e.message, “Expression:”, expression);
return NaN; // Indicate failure
}
}

function integrate(funcStr, variable, start, end, steps = 1000) {
var h = (end – start) / steps;
var sum = 0;
for (var i = 0; i < steps; i++) {
var x0 = start + i * h;
var x1 = start + (i + 1) * h;
var mid = (x0 + x1) / 2;

var val1 = evaluateExpression(funcStr.replace(new RegExp(variable, 'g'), x0.toString()));
var val2 = evaluateExpression(funcStr.replace(new RegExp(variable, 'g'), x1.toString()));

if (isNaN(val1) || isNaN(val2)) return NaN;

sum += (val1 + val2) * h / 2;
}
return sum;
}

// — Calculator Logic —
function updateInputLabels() {
var type = document.getElementById('equationType').value;
currentEquationType = type;

if (type === 'simple') {
document.getElementById('inputGroupFx').style.display = 'flex';
document.getElementById('inputGroupGy').style.display = 'flex';
document.getElementById('inputGroupM').style.display = 'none';
document.getElementById('inputGroupN').style.display = 'none';
document.getElementById('labelFx').innerText = 'Function f(x):';
document.getElementById('labelGy').innerText = 'Function g(y):';
} else { // advanced
document.getElementById('inputGroupFx').style.display = 'none';
document.getElementById('inputGroupGy').style.display = 'none';
document.getElementById('inputGroupM').style.display = 'flex';
document.getElementById('inputGroupN').style.display = 'flex';
document.getElementById('labelM').innerText = 'Function M(x):';
document.getElementById('labelN').innerText = 'Function N(y):';
}
clearResults();
}

function clearErrorMessages() {
document.getElementById('errorFx').innerText = '';
document.getElementById('errorGy').innerText = '';
document.getElementById('errorM').innerText = '';
document.getElementById('errorN').innerText = '';
document.getElementById('errorInitialX').innerText = '';
document.getElementById('errorInitialY').innerText = '';
document.getElementById('errorXValue').innerText = '';
}

function clearResults() {
document.getElementById('primaryResult').innerText = '–';
document.getElementById('intermediate1').innerHTML = 'Integral of f(x): –‘;
document.getElementById(‘intermediate2’).innerHTML = ‘Integral of g(y): –‘;
document.getElementById(‘intermediate3’).innerHTML = ‘Constant C: –‘;
clearChart();
}

function validateInputs() {
var valid = true;
var inputs = {
fx: document.getElementById(‘fx’).value,
gy: document.getElementById(‘gy’).value,
m: document.getElementById(‘m’).value,
n: document.getElementById(‘n’).value,
initialX: document.getElementById(‘initialX’).value,
initialY: document.getElementById(‘initialY’).value,
xValue: document.getElementById(‘xValue’).value
};

clearErrorMessages();

var initialX = parseFloat(inputs.initialX);
var initialY = parseFloat(inputs.initialY);
var xValue = parseFloat(inputs.xValue);

if (isNaN(initialX)) {
document.getElementById(‘errorInitialX’).innerText = ‘Please enter a valid number for x₀.’;
valid = false;
}
if (isNaN(initialY)) {
document.getElementById(‘errorInitialY’).innerText = ‘Please enter a valid number for y₀.’;
valid = false;
}
if (isNaN(xValue)) {
document.getElementById(‘errorXValue’).innerText = ‘Please enter a valid number for the evaluation x.’;
valid = false;
}

if (currentEquationType === ‘simple’) {
if (!inputs.fx) {
document.getElementById(‘errorFx’).innerText = ‘f(x) cannot be empty.’;
valid = false;
}
if (!inputs.gy) {
document.getElementById(‘errorGy’).innerText = ‘g(y) cannot be empty.’;
valid = false;
}
// Check if g(y) is zero at initialY
var gyVal = evaluateExpression(inputs.gy, null, initialY);
if (gyVal === 0) {
document.getElementById(‘errorGy’).innerText = ‘g(y) cannot be zero at y₀ for this method.’;
valid = false;
}
} else { // advanced
if (!inputs.m) {
document.getElementById(‘errorM’).innerText = ‘M(x) cannot be empty.’;
valid = false;
}
if (!inputs.n) {
document.getElementById(‘errorN’).innerText = ‘N(y) cannot be empty.’;
valid = false;
}
}

return valid ? { inputs: inputs, initialX: initialX, initialY: initialY, xValue: xValue } : false;
}

function calculateSolution() {
var validationData = validateInputs();
if (!validationData) return;

var inputs = validationData.inputs;
var initialX = validationData.initialX;
var initialY = validationData.initialY;
var xValue = validationData.xValue;

var fxStr = inputs.fx;
var gyStr = inputs.gy;
var mStr = inputs.m;
var nStr = inputs.n;

var integralFxResult = NaN;
var integralGyResult = NaN;
var constantC = NaN;
var finalY = NaN;

try {
if (currentEquationType === ‘simple’) {
// Integrate f(x) from initialX to xValue
integralFxResult = integrate(fxStr, ‘x’, initialX, xValue);

// Integrate 1/g(y) from initialY to y
// This requires an iterative approach or implicit function solving.
// For simplicity, we’ll calculate C and then try to find y.
// Let’s calculate the integral of 1/g(y) from initialY up to some arbitrary y_test
// and the integral of f(x) from initialX up to xValue.
// G(y) = F(x) + C
// G(initialY) = F(initialX) + C => C = G(initialY) – F(initialX)

// Calculate F(initialX)
var F_initialX = integrate(fxStr, ‘x’, 0, initialX); // Integrate from 0 for reference

// Calculate G(initialY) – requires integrating 1/g(y)
// We need a way to represent the integral of 1/g(y) symbolically or numerically.
// For this calculator, we’ll approximate the integral of 1/g(y) from initialY to a test y.
// A simpler approach for the calculator: Calculate C using the initial condition.
// ∫(1/g(y)) dy = ∫f(x) dx + C
// Let’s evaluate the integral of f(x) from initialX to xValue.
var integral_fx_part = integrate(fxStr, ‘x’, initialX, xValue);

// We need to find y such that ∫(1/g(y)) dy evaluated from initialY to y equals integral_fx_part.
// This is hard to do directly without symbolic integration or numerical root finding.

// Let’s simplify: Calculate C based on initial conditions.
// We need a function G(y) = ∫(1/g(y)) dy.
// Let’s approximate G(y) numerically.
var G_initialY = integrate(‘1/’ + gyStr, ‘y’, 0, initialY); // Integrate from 0 for reference
var F_xValue = integrate(fxStr, ‘x’, 0, xValue); // Integrate from 0 for reference

if (isNaN(G_initialY) || isNaN(F_xValue)) {
throw new Error(“Integration failed for initial conditions.”);
}

constantC = G_initialY – F_xValue;

// Now find y such that G(y) = F(xValue) + C
var targetIntegralValue = F_xValue + constantC;

// Numerical root finding for y
var y_low = initialY – 100; // Search range, adjust as needed
var y_high = initialY + 100;
var y_step = (y_high – y_low) / 2000;
var foundY = NaN;

for (var y_test = y_low; y_test <= y_high; y_test += y_step) {
var G_y_test = integrate('1/' + gyStr, 'y', 0, y_test);
if (!isNaN(G_y_test) && Math.abs(G_y_test – targetIntegralValue) < 1e-6) {
foundY = y_test;
break;
}
}
finalY = foundY;

// Update intermediate results display
document.getElementById('intermediate1').innerHTML = 'Integral of f(x) from x₀ to x: ‘ + integral_fx_part.toFixed(4);
document.getElementById(‘intermediate2’).innerHTML = ‘Integral of 1/g(y) from y₀ to y: ‘ + (targetIntegralValue – constantC).toFixed(4); // This is G(y) – G(y0)
document.getElementById(‘intermediate3’).innerHTML = ‘Constant C: ‘ + constantC.toFixed(4);

} else { // advanced: M(x)dx + N(y)dy = 0
var integralM = integrate(mStr, ‘x’, initialX, xValue); // Assuming x is the variable for M
var integralN = integrate(nStr, ‘y’, initialY, initialY); // Integrate N(y) from initialY to itself (0) for C calculation
// This form is simpler: ∫M(x)dx + ∫N(y)dy = C
// We need to find y such that ∫M(x)dx evaluated from initialX to xValue + ∫N(y)dy evaluated from initialY to y = 0 (or some constant if M/N are not exact)
// Assuming M(x)dx + N(y)dy = 0 is exact or separable in a way that leads to ∫M(x)dx + ∫N(y)dy = C
var integral_M_part = integrate(mStr, ‘x’, 0, xValue); // Integrate M(x) from 0 to xValue
var integral_N_part = integrate(nStr, ‘y’, 0, initialY); // Integrate N(y) from 0 to initialY

if (isNaN(integral_M_part) || isNaN(integral_N_part)) {
throw new Error(“Integration failed for M(x) or N(y).”);
}

constantC = -(integral_M_part + integral_N_part); // C = – (Integral M(x)dx + Integral N(y)dy) evaluated at initial conditions

// Find y such that Integral M(x)dx (from 0 to xValue) + Integral N(y)dy (from 0 to y) = 0
var targetIntegralValue = 0; // For Mdx + Ndy = 0, the constant is often implicitly 0 after rearrangement.
// Or, more accurately, ∫M(x)dx + ∫N(y)dy = C.
// Let’s use the form: ∫M(x)dx evaluated from initialX to xValue + ∫N(y)dy evaluated from initialY to y = 0
var integral_M_at_xValue = integrate(mStr, ‘x’, initialX, xValue);
var integral_N_at_initialY = integrate(nStr, ‘y’, initialY, initialY); // This is 0

if (isNaN(integral_M_at_xValue)) throw new Error(“Integration failed for M(x).”);

var target_N_integral = -integral_M_at_xValue; // We need ∫N(y)dy from initialY to y = -∫M(x)dx from initialX to xValue

// Numerical root finding for y
var y_low = initialY – 100;
var y_high = initialY + 100;
var y_step = (y_high – y_low) / 2000;
var foundY = NaN;

for (var y_test = y_low; y_test <= y_high; y_test += y_step) {
var N_integral_y_test = integrate(nStr, 'y', initialY, y_test);
if (!isNaN(N_integral_y_test) && Math.abs(N_integral_y_test – target_N_integral) < 1e-6) {
foundY = y_test;
break;
}
}
finalY = foundY;

document.getElementById('intermediate1').innerHTML = 'Integral of M(x) from x₀ to x: ‘ + integral_M_at_xValue.toFixed(4);
document.getElementById(‘intermediate2’).innerHTML = ‘Integral of N(y) from y₀ to y: ‘ + target_N_integral.toFixed(4); // This is the target value for the N integral
document.getElementById(‘intermediate3’).innerHTML = ‘Constant C: ‘ + constantC.toFixed(4); // This C might be less directly interpretable here.
}

if (isNaN(finalY)) {
throw new Error(“Could not find a valid solution for y.”);
}

document.getElementById(‘primaryResult’).innerText = finalY.toFixed(6);

// Update chart
updateChart(initialX, initialY, xValue, finalY, fxStr, gyStr, mStr, nStr);

} catch (e) {
console.error(“Calculation error:”, e.message);
document.getElementById(‘primaryResult’).innerText = ‘Error’;
document.getElementById(‘intermediate1’).innerHTML = ‘Integral of f(x): Error’;
document.getElementById(‘intermediate2’).innerHTML = ‘Integral of g(y): Error’;
document.getElementById(‘intermediate3’).innerHTML = ‘Constant C: Error’;
alert(“Calculation failed: ” + e.message + “\nEnsure functions are valid and separable.”);
}
}

function resetCalculator() {
document.getElementById(‘equationType’).value = ‘simple’;
document.getElementById(‘fx’).value = ‘x’;
document.getElementById(‘gy’).value = ‘y’;
document.getElementById(‘m’).value = ‘x’;
document.getElementById(‘n’).value = ‘y’;
document.getElementById(‘initialX’).value = ‘0’;
document.getElementById(‘initialY’).value = ‘1’;
document.getElementById(‘xValue’).value = ‘2’;

updateInputLabels();
clearErrorMessages();
clearResults();
document.getElementById(‘primaryResult’).innerText = ‘–‘;
document.getElementById(‘intermediate1’).innerHTML = ‘Integral of f(x): –‘;
document.getElementById(‘intermediate2’).innerHTML = ‘Integral of g(y): –‘;
document.getElementById(‘intermediate3’).innerHTML = ‘Constant C: –‘;
}

function copyResults() {
var primaryResult = document.getElementById(‘primaryResult’).innerText;
var intermediate1 = document.getElementById(‘intermediate1’).innerText.replace(‘Integral of f(x):’, ‘Integral f(x):’).replace(‘Integral of M(x) from x₀ to x:’, ‘Integral M(x):’).replace(‘Integral of 1/g(y) from y₀ to y:’, ‘Integral 1/g(y):’).replace(‘Integral of N(y) from y₀ to y:’, ‘Integral N(y):’);
var intermediate2 = document.getElementById(‘intermediate2’).innerText;
var intermediate3 = document.getElementById(‘intermediate3’).innerText;

var equationType = document.getElementById(‘equationType’).value;
var fx = document.getElementById(‘fx’).value;
var gy = document.getElementById(‘gy’).value;
var m = document.getElementById(‘m’).value;
var n = document.getElementById(‘n’).value;
var initialX = document.getElementById(‘initialX’).value;
var initialY = document.getElementById(‘initialY’).value;
var xValue = document.getElementById(‘xValue’).value;

var assumptions = “Equation Type: ” + equationType + “\n”;
if (equationType === ‘simple’) {
assumptions += “f(x): ” + fx + “\n”;
assumptions += “g(y): ” + gy + “\n”;
} else {
assumptions += “M(x): ” + m + “\n”;
assumptions += “N(y): ” + n + “\n”;
}
assumptions += “Initial x (x₀): ” + initialX + “\n”;
assumptions += “Initial y (y₀): ” + initialY + “\n”;
assumptions += “Evaluate at x: ” + xValue + “\n”;

var textToCopy = “Separation of Variables Calculation Results:\n\n”;
textToCopy += “Primary Result (y at x=” + xValue + “): ” + primaryResult + “\n\n”;
textToCopy += “Intermediate Values:\n”;
textToCopy += “- ” + intermediate1 + “\n”;
textToCopy += “- ” + intermediate2 + “\n”;
textToCopy += “- ” + intermediate3 + “\n\n”;
textToCopy += “Assumptions:\n” + assumptions;

navigator.clipboard.writeText(textToCopy).then(function() {
alert(‘Results copied to clipboard!’);
}, function(err) {
alert(‘Failed to copy results. Please copy manually.’);
console.error(‘Clipboard copy failed: ‘, err);
});
}

// — Charting Logic —
var solutionChart;
var chartData = {
labels: [],
datasets: [{
label: ‘Solution Curve’,
data: [],
borderColor: ‘var(–primary-color)’,
backgroundColor: ‘rgba(0, 74, 153, 0.1)’,
fill: false,
tension: 0.1,
pointRadius: 0
}, {
label: ‘Initial Point’,
data: [],
borderColor: ‘var(–success-color)’,
backgroundColor: ‘var(–success-color)’,
pointRadius: 5,
showLine: false
}]
};

function clearChart() {
if (solutionChart) {
solutionChart.destroy();
solutionChart = null;
}
chartData.labels = [];
chartData.datasets[0].data = [];
chartData.datasets[1].data = [];
}

function updateChart(initialX, initialY, targetX, targetY, fxStr, gyStr, mStr, nStr) {
clearChart();
var ctx = document.getElementById(‘solutionChart’).getContext(‘2d’);

// Generate points for the solution curve
var numPoints = 50;
var xMin = Math.min(initialX, targetX) – Math.abs(initialX – targetX) * 0.5;
var xMax = Math.max(initialX, targetX) + Math.abs(initialX – targetX) * 0.5;
var xStep = (xMax – xMin) / numPoints;

var points = [];
for (var i = 0; i <= numPoints; i++) {
var currentX = xMin + i * xStep;
var currentY = NaN;

// Try to find y for currentX using the calculated C and the appropriate integral form
try {
if (currentEquationType === 'simple') {
var F_currentX = integrate(fxStr, 'x', 0, currentX);
var G_initialY = integrate('1/' + gyStr, 'y', 0, initialY); // Recalculate G(initialY)
var F_initialX = integrate(fxStr, 'x', 0, initialX); // Recalculate F(initialX)

if (isNaN(F_currentX) || isNaN(G_initialY) || isNaN(F_initialX)) continue;

var targetIntegralValue = F_currentX + (G_initialY – F_initialX);

// Numerical root finding for y
var y_low = initialY – 100;
var y_high = initialY + 100;
var y_step = (y_high – y_low) / 2000;
var foundY = NaN;

for (var y_test = y_low; y_test <= y_high; y_test += y_step) {
var G_y_test = integrate('1/' + gyStr, 'y', 0, y_test);
if (!isNaN(G_y_test) && Math.abs(G_y_test – targetIntegralValue) < 1e-6) {
foundY = y_test;
break;
}
}
currentY = foundY;

} else { // advanced: M(x)dx + N(y)dy = 0
var integral_M_currentX = integrate(mStr, 'x', initialX, currentX);
if (isNaN(integral_M_currentX)) continue;

var target_N_integral = -integral_M_currentX;

// Numerical root finding for y
var y_low = initialY – 100;
var y_high = initialY + 100;
var y_step = (y_high – y_low) / 2000;
var foundY = NaN;

for (var y_test = y_low; y_test <= y_high; y_test += y_step) {
var N_integral_y_test = integrate(nStr, 'y', initialY, y_test);
if (!isNaN(N_integral_y_test) && Math.abs(N_integral_y_test – target_N_integral) p.x.toFixed(2));
chartData.datasets[0].data = points.map(p => ({ x: p.x, y: p.y }));
chartData.datasets[1].data = [{ x: initialX, y: initialY }]; // Initial point

// Add the target point if it’s not already the last point
if (points.length === 0 || Math.abs(points[points.length – 1].x – targetX) > 1e-6) {
chartData.datasets[0].data.push({ x: targetX, y: targetY });
chartData.labels.push(targetX.toFixed(2));
}

solutionChart = new Chart(ctx, {
type: ‘line’,
data: chartData,
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
x: {
title: { display: true, text: ‘x’ },
type: ‘linear’,
position: ‘bottom’
},
y: {
title: { display: true, text: ‘y’ }
}
},
plugins: {
legend: { display: false }, // Use custom legend
tooltip: {
callbacks: {
label: function(context) {
var label = context.dataset.label || ”;
if (label) {
label += ‘: ‘;
}
if (context.parsed.x !== null) {
label += ‘(‘ + context.parsed.x.toFixed(3) + ‘, ‘ + context.parsed.y.toFixed(3) + ‘)’;
}
return label;
}
}
}
}
}
});
}

// — Initial Setup —
window.onload = function() {
updateInputLabels();
resetCalculator(); // Set default values and clear results
// Initialize chart canvas
var canvas = document.getElementById(‘solutionChart’);
var ctx = canvas.getContext(‘2d’);
// Create an empty chart initially
solutionChart = new Chart(ctx, {
type: ‘line’,
data: { datasets: [] }, // Empty data initially
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
x: { title: { display: true, text: ‘x’ } },
y: { title: { display: true, text: ‘y’ } }
},
plugins: { legend: { display: false } }
}
});
};