Differential Equation Solver
ODE Solver
Solve ordinary differential equations (ODEs) with step-by-step solutions. Supports first order, second order, linear and separable equations.
Differential Equation Solution
General Solution
General Solution:
Step-by-Step Solution:
Equation Analysis:
Solution Visualization:
The solution satisfies the differential equation and initial conditions.
What are Differential Equations?
Differential Equations are mathematical equations that relate a function with its derivatives. They describe how quantities change over time or space and are fundamental in modeling physical systems, engineering processes, biological growth, economic trends, and many other natural phenomena.
Types of Differential Equations
Ordinary Differential Equations (ODEs)
One independent variable
Most common type
Partial Differential Equations (PDEs)
Multiple independent variables
Heat equation, wave equation
Linear Differential Equations
Linear in y and derivatives
Superposition applies
Nonlinear Differential Equations
Nonlinear terms
More complex behavior
Solution Methods for ODEs
1. Separation of Variables
For equations of the form dy/dx = g(x)h(y):
dy/dx = g(x)h(y)
∫(1/h(y)) dy = ∫g(x) dx
Solve for y
2. Integrating Factor Method
For linear first order equations y' + P(x)y = Q(x):
μ(x) = exp(∫P(x) dx)
Multiply: μ(x)y' + μ(x)P(x)y = μ(x)Q(x)
d/dx[μ(x)y] = μ(x)Q(x)
3. Characteristic Equation Method
For constant coefficient linear equations:
ay'' + by' + cy = 0
Characteristic equation: ar² + br + c = 0
Roots determine solution form
Real-World Applications
Physics & Engineering
- Newton's Second Law: F = ma (second order ODE)
- RC Circuits: Capacitor charging/discharging (first order)
- Spring-mass systems: Harmonic oscillator (second order)
- Heat conduction: Fourier's law (partial differential equation)
Biology & Medicine
- Population growth: Exponential and logistic models
- Epidemiology: SIR model for disease spread
- Pharmacokinetics: Drug concentration over time
- Neural networks: Hodgkin-Huxley equations
Economics & Finance
- Economic growth: Solow-Swan model
- Option pricing: Black-Scholes equation (PDE)
- Interest models: Continuous compounding
- Market dynamics: Price evolution models
Everyday Examples
- Cooling/heating: Newton's law of cooling
- Radioactive decay: First order decay equation
- Compound interest: Continuous growth model
- Motion under gravity: Projectile trajectories
Common ODE Examples and Solutions
| ODE Type | Equation | General Solution | Application |
|---|---|---|---|
| Exponential Growth | dy/dt = ky | y = y₀e^(kt) | Population growth, radioactive decay |
| Harmonic Oscillator | d²y/dt² + ω²y = 0 | y = A cos(ωt + φ) | Spring-mass systems, pendulums |
| Logistic Growth | dy/dt = ry(1-y/K) | y = K/(1 + Ce^(-rt)) | Population with carrying capacity |
| Newton's Cooling | dT/dt = -k(T - T_env) | T = T_env + (T₀-T_env)e^(-kt) | Temperature change |
Solution Characteristics
| Property | Description | Example ODE | Implication |
|---|---|---|---|
| Order | Highest derivative present | d²y/dx² + y = 0 (2nd order) | Number of initial conditions needed |
| Linearity | Linear in y and derivatives | y'' + xy' + y = 0 (linear) | Superposition principle applies |
| Homogeneity | Right side = 0 | y' + P(x)y = 0 (homogeneous) | Solution space is vector space |
| Autonomy | No explicit x dependence | dy/dx = f(y) (autonomous) | Time-invariant system |
Step-by-Step Solution Process
Example 1: dy/dx = x + y (First Order Linear)
- Write in standard form: y' - y = x
- Identify P(x) = -1, Q(x) = x
- Find integrating factor: μ(x) = e^(∫-1 dx) = e^(-x)
- Multiply: e^(-x)y' - e^(-x)y = xe^(-x)
- Left side is derivative: d/dx[e^(-x)y] = xe^(-x)
- Integrate both sides: e^(-x)y = ∫xe^(-x)dx
- Solve integral: ∫xe^(-x)dx = -xe^(-x) - e^(-x) + C
- Multiply by e^x: y = -x - 1 + Ce^x
- General solution: y = Ce^x - x - 1
Example 2: d²y/dx² + 4y = 0 (Second Order Constant Coefficient)
- Characteristic equation: r² + 4 = 0
- Solve: r = ±2i (complex roots)
- General solution: y = C₁ cos(2x) + C₂ sin(2x)
- With initial conditions: y(0)=1, y'(0)=0
- Apply y(0)=1: 1 = C₁ cos(0) + C₂ sin(0) → C₁ = 1
- Differentiate: y' = -2C₁ sin(2x) + 2C₂ cos(2x)
- Apply y'(0)=0: 0 = -2(1) sin(0) + 2C₂ cos(0) → C₂ = 0
- Particular solution: y = cos(2x)
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between ordinary and partial differential equations?
A: Ordinary Differential Equations (ODEs) involve derivatives with respect to one independent variable (usually time or position). Partial Differential Equations (PDEs) involve derivatives with respect to multiple independent variables (like time and space). ODEs describe single-variable systems, PDEs describe multi-variable systems.
Q: How many initial conditions do I need for an nth order ODE?
A: For an nth order ODE, you need n initial conditions to determine a unique solution. For example: 1st order needs y(x₀), 2nd order needs y(x₀) and y'(x₀), 3rd order needs y(x₀), y'(x₀), and y''(x₀).
Q: What does it mean if an ODE is "linear"?
A: A linear ODE is linear in the dependent variable and its derivatives. Form: aₙ(x)y⁽ⁿ⁾ + ... + a₁(x)y' + a₀(x)y = g(x). No products or nonlinear functions of y or its derivatives. Linear ODEs obey superposition: if y₁ and y₂ are solutions, then c₁y₁ + c₂y₂ is also a solution.
Q: When should I use numerical methods vs analytical solutions?
A: Use analytical solutions when they exist and are practical (gives formula). Use numerical methods (Euler, Runge-Kutta) when: 1) No analytical solution exists, 2) Solution is too complex, 3) Need numerical values, 4) Dealing with systems of ODEs. Most real-world problems require numerical methods.
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