Tutorial 10: First-Order Linear Differential Equations

Tutorial problems covering first-order linear differential equations and integrating factor method.

Sections

Standard Form Linear DEs (Problems 1-4)

  • Identifying linear first-order DEs
  • Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$
  • Finding integrating factors

Integrating Factor Method (Problems 5-8)

  • Compute $\mu(x) = e^{\int P(x),dx}$
  • Multiply through by integrating factor
  • Solve resulting exact equation

Applications (Problems 9-12)

  • Circuit problems
  • Mixing with variable rates
  • Physics applications

Integrating Factor Method

For $\frac{dy}{dx} + P(x)y = Q(x)$:

  1. Find integrating factor: $\mu(x) = e^{\int P(x),dx}$
  2. Multiply equation: $\mu(x)\frac{dy}{dx} + \mu(x)P(x)y = \mu(x)Q(x)$
  3. Left side becomes: $\frac{d}{dx}[\mu(x)y] = \mu(x)Q(x)$
  4. Integrate: $\mu(x)y = \int \mu(x)Q(x),dx$
  5. Solve for y: $y = \frac{1}{\mu(x)}\int \mu(x)Q(x),dx$

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