First-Order Differential Equation Calculator
Solve simple first-order differential equation models such as dy/dx = ky and dy/dx + py = q with an initial condition.
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What this calculator teaches
A first-order differential equation relates a function to its first derivative. It describes change over time or another independent variable.
This calculator covers two foundational models: proportional growth/decay and a constant-input linear equation.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Proportional model | dy/dx = ky → y = y₀e^{kx} | Growth if k>0, decay if k<0. |
| Linear constant model | dy/dx + py = q | Solution approaches q/p when p>0. |
| Initial condition | y(0)=y₀ | Selects one solution from the family. |
Step-by-Step Examples
- dy/dx = 0.4y.
- Initial value y(0)=5.
- Solution is y = 5e^{0.4x}.
- dy/dx + 2y = 10.
- Equilibrium is q/p = 5.
- Solution moves toward 5.
Where students use this
Common Mistakes to Avoid
The differential equation gives a family of solutions until y(0) is used.
dy/dx=ky uses e^{kx}; dy/dx + py = q uses e^{-px}.
Many first-order equations require other methods or numerical solving.
Frequently Asked Questions
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Formula Explorer connections
Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.