Differential Equations

A differential equation is an equation involving a function and its derivatives. It describes how the function changes, and solving it means finding the function that satisfies the equation.

Basic Concepts

Order and Degree

The order of a differential equation is the highest derivative order present.

The degree is the power of the highest order derivative (if the equation is polynomial in derivatives).

Types of Differential Equations

Ordinary Differential Equations (ODEs) involve derivatives with respect to one variable.

Partial Differential Equations (PDEs) involve partial derivatives with respect to multiple variables.

Linear vs Nonlinear

Linear differential equations have the function and its derivatives appearing linearly.

Nonlinear differential equations involve nonlinear terms of the function or its derivatives.

Simple First-Order ODE Example

Equation: dy/dx = 3x^2

Solution: Integrate both sides w.r.t. x:

  • ∫ dy = ∫ 3x² dx
  • y = x³ + C, where C is the constant of integration.

Separable Differential Equations

A differential equation is separable if it can be written as:

dy/dx = g(x) h(y)

Rewrite as:

dy / h(y) = g(x) dx

Then integrate both sides.

Equation: dy/dx = xy

Solution:

  • Rewrite: (1/y) dy = x dx
  • Integrate both sides: ∫(1/y) dy = ∫ x dx
  • ln|y| = x²/2 + C
  • y = ± e^(x²/2 + C) = Ae^(x²/2)

Step-by-Step: Solving a Separable ODE

  1. Rewrite the equation in separable form: dy/dx = g(x)h(y).
  2. Separate variables: dy/h(y) = g(x) dx.
  3. Integrate both sides.
  4. Solve the resulting equation for y, if possible.
  5. Include constant of integration.

Practice Problems

1. Solve: dy/dx = 2x

2. Solve: dy/dx = yx

3. Solve: dy/dx = y^2