MATH1012 - Lecture 27

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Differential Equations

• A differential equation is any equation that involves a derivative
  • Need to find all functions that satisfy a given differential equation
  • i.e. solve the differential equation
• The order of a differential equation is the order of the highest derivative which appears within the equation
  • e.g. an equation with $\frac{d^{2}y}{d{x}^2}$ is a 2nd order derivative

Separation of Variables

• A first order differential equation is said to be separable if it can be written in the form $\frac{dy}{dx}=F(x)G(y)$
  • We can solve such differential equations by a process called separation of variables
• Rewrite + integrate both sides: $\frac{dy}{dx}=F(x)G(y)\to \int \frac{1}{G(y)}dy=\int F(x)dx$

First-Order Linear Differential Equations

• First-order linear differential equations are equations that can be written in the form:
  $\frac{dy}{dx}+f(x)y=g(x)$

• To solve, we multiply both sides by an integrating factor: $I(x)=e^{\int f(x)dx}$

  • The derivative of this factor is: $f(x)I(x)$

• We get: $I(x)\frac{dy}{dx}+I(x)f(x)y=I(x)g(x)$ which simplifies down to:
  $\frac{d}{dx}\{I(x)y\}=I(x)g(x)$

• From here, you can integrate both sides $\int \frac{d}{dx}\{I(x)y\}dt=\int I(x)g(x)dt$: $I(x)y=\int I(x)g(x)dt$

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MATH1012 - Lecture 28
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