This tool solves the fundamental first-order linear differential equation, $\frac{dy}{dt} = k y$, which models continuous exponential growth or decay. It allows you to find any missing variable (Initial Value, Rate, Time, or Final Value) given the other three.
Differential Equations Solver (Exponential Growth)
The calculated missing variable is:
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Differential Equations Solver Formula:
Variables:
- Final Value ($F$): The final amount, population, or quantity after $T$ time periods.
- Initial Value ($P$): The starting amount, population, or principal at time $t=0$.
- Growth/Decay Rate ($R$): The continuous rate of change (e.g., interest rate, growth factor) expressed as a decimal (e.g., 5% $\rightarrow$ 0.05).
- Time Periods ($T$): The duration over which the growth or decay occurs. Must be positive.
- $e$: Euler’s number (approximately 2.71828), the base of the natural logarithm.
Related Calculators:
- Continuous Compounding Calculator
- Radioactive Decay Half-Life Calculator
- Logistic Growth Model Predictor
- First-Order ODE Homogeneous Solver
What is Differential Equations Calculator?
A differential equations calculator, in this context, is a tool designed to solve simple, commonly encountered differential equations that have closed-form solutions. The most fundamental of these is the **Exponential Growth/Decay Model** (also known as the Malthusian Model), which is central to continuous finance, biology, and physics.
The underlying equation, $\frac{dy}{dt} = k y$, states that the rate of change of a quantity ($y$) is proportional to the quantity itself. This is solved analytically to yield the formula $F = P e^{RT}$. This calculator uses that analytic solution to quickly determine one unknown parameter given the others, saving the user from manual logarithmic or exponential calculations.
How to Calculate Differential Equations (Example):
Suppose a bacterial culture starts with 500 cells and grows continuously at a rate of 12% ($R=0.12$) over 5 hours ($T=5$). What is the final population ($F$)?
- Identify Known Variables: Initial Value ($P$) = 500, Rate ($R$) = 0.12, Time ($T$) = 5.
- Select Formula: Since $F$ is unknown, use the formula $F = P e^{RT}$.
- Substitute Values: $F = 500 \times e^{(0.12 \times 5)}$.
- Calculate Exponent: $0.12 \times 5 = 0.6$. The equation becomes $F = 500 \times e^{0.6}$.
- Evaluate Exponential Term: $e^{0.6} \approx 1.822119$.
- Determine Final Value: $F = 500 \times 1.822119 \approx 911.06$.
- Result: The final population after 5 hours is approximately 911.06 cells.
Frequently Asked Questions (FAQ):
- What is a differential equation?
- A differential equation is an equation that relates one or more functions and their derivatives. It connects a function with its rate of change, making it a powerful tool for modeling dynamic processes in the real world.
- Is the Exponential Growth Model a good fit for all growth?
- No. While it is excellent for early-stage, unrestricted growth (like simple interest or initial population growth), real-world systems often have limitations. For scenarios with carrying capacity, the **Logistic Differential Equation** is a more accurate model.
- What does the rate $R$ represent in this context?
- The rate $R$ represents the *continuous compounding* factor. If $R$ is positive, it signifies growth; if $R$ is negative, it signifies decay (e.g., radioactive decay). This rate is derived directly from the differential equation’s constant of proportionality ($k$).
- Why do I need to input at least three variables?
- The equation $F = P e^{RT}$ has four variables. To solve for a unique unknown, you need three known values. If you input all four, the calculator will check for mathematical consistency.