First Order Linear DEQs

Form

$$a_1(x)\frac{dy}{dx}+a_0(x)y=b(x)\text{ where }{a}_1(x)\ne 0$$

Solving

General Case

1. Write in form $\frac{dy}{dx}+P(x)y=Q(x)$
2. Find the integrating factor $\mu (x)=e^{\int P(x)dx}$
   • We just need an antiderivative, not all, so set $C$ to 0
3. Multiply (1) by $\mu (x)$
4. Apply the reverse power rule to get $\frac{d}{dx}[\mu (x)y]=\mu (x)Q(x)$
5. Integrate both sides
6. Solve for y

Case 1

$$\text{ if }{a}_0(x)=0$$ $$y(x)=\int \frac{b(x)}{a_1(x)}dx+C$$

Case 2

$$\text{if }{a}_0(x)=a_1^{\prime }(x)$$ $$y(x)=\frac{\int b(x)dx+C}{a_1(x)}$$