Tutorial 7: Differential Equations

Tutorial problems covering separable differential equations and applications.

Sections

Solving Separable DEs (Problems 1-4)

  • Separating variables
  • Integrating both sides
  • Finding general solutions
  • Particular solutions from initial conditions

Modeling Problems (Problems 5-8)

  • Exponential growth/decay
  • Population models
  • Newton's Law of Cooling
  • Mixing problems

Advanced Applications (Problems 9-12)

  • Logistic growth
  • Complex modeling scenarios
  • Real-world applications

Standard Solution Process

  1. Separate: $\frac{dy}{g(y)} = f(x),dx$
  2. Integrate: $\int \frac{dy}{g(y)} = \int f(x),dx$
  3. Solve: Express $y$ in terms of $x$ (if possible)
  4. Apply IC: Use initial condition to find constant

Common Models

Exponential Growth/Decay: $$\frac{dy}{dt} = ky \implies y = y_0e^{kt}$$

Newton's Law of Cooling: $$\frac{dT}{dt} = k(T - T_s)$$

Logistic Growth: $$\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$$

Links