Differential Equations • Topic 3 of 3

Linear Differential Equations

A first-order linear differential equation has the standard form

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

where $P$ and $Q$ are functions of $x$ only. It is solved using an integrating factor.

The integrating factor (IF)

$$\text{IF}=e^{\int P\,dx}.$$

Multiplying through by the IF turns the left side into the derivative of (IF $\times y$). The solution is then

$$y\cdot(\text{IF})=\int Q\cdot(\text{IF})\,dx + C.$$

Method

  1. Put the equation in standard form and read off $P$ and $Q$.
  2. Compute IF $=e^{\int P\,dx}$.
  3. Write $y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C$ and integrate.
Solved Examples
1
Worked Example
Find the integrating factor of $\dfrac{dy}{dx}+\dfrac{y}{x}=x$.
Solution

$P=\dfrac1x$, so IF $=e^{\int \frac1x dx}=e^{\ln x}=x.$

2
Worked Example
Solve $\dfrac{dy}{dx}+\dfrac{y}{x}=x$.
Solution

IF $=x$. Then $y\cdot x=\int x\cdot x\,dx=\int x^2 dx=\dfrac{x^3}{3}+C$. So $y=\dfrac{x^2}{3}+\dfrac{C}{x}.$

3
Worked Example
Find the IF of $\dfrac{dy}{dx}+2y=e^{x}$.
Solution

$P=2$, IF $=e^{\int 2\,dx}=e^{2x}.$

4
Worked Example
Solve $\dfrac{dy}{dx}+2y=e^{x}$.
Solution

IF $=e^{2x}$. $y e^{2x}=\int e^{x}e^{2x}dx=\int e^{3x}dx=\dfrac{e^{3x}}{3}+C$. So $y=\dfrac{e^{x}}{3}+Ce^{-2x}.$

Key Points

  • Standard linear form: $\dfrac{dy}{dx}+P(x)y=Q(x)$.
  • Integrating factor IF $=e^{\int P\,dx}$.
  • Solution: $y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx+C$.
  • Always put the equation in standard form before reading $P$ and $Q$.
Quick Check — 4 MCQs
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Q1.The integrating factor of $\dfrac{dy}{dx}+\dfrac{y}{x}=x$ is:
Explanation: $e^{\int (1/x)dx}=e^{\ln x}=x$.
Q2.For $\dfrac{dy}{dx}+2y=e^x$, the IF is:
Explanation: $e^{\int 2dx}=e^{2x}$.
Q3.A linear first-order DE has the form:
Explanation: Standard linear form.
Q4.After multiplying by the IF, the left side becomes:
Explanation: That is the whole point of the IF.
Practice more MCQs → 📝 Assignment · 35 marks