# First-Order Linear Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 2.6 – Linear Equations

## Expected Educational Results

• Objective 6–1: I can identify if a first-order ordinary differential equation is linear.

• Objective 6–2: I rewrite a first-order linear ordinary differential equation in standard form.

• Objective 6–3: I can solve a first-order linear ordinary differential equation.

• Objective 6–4: I can solve a first-order linear initial value problem.

• Objective 6–5: I identify missing solutions to a first-order linear ordinary differential equation.

## Linear Equations

### Method for Solving First-Order ODEs

1. Identify if the DE is a linear first-order DE;

2. Rewrite the DE in standard form, $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$\displaystyle \frac{dy}{dx}+P(x)y=Q(x)$;

3. Compute the integrating factor: $\mu \left(x\right)={e}^{\int P\left(x\right)dx}$$\displaystyle \mu (x)=e^{\int{P(x)dx}}$ [no $+C$$+C$ needed here];

4. Multiply both sides of the DE in standard form by: $\mu \left(x\right)$$\mu (x)$ to obtain $\mu \left(x\right)\frac{dy}{dx}+\mu \left(x\right)P\left(x\right)y=\mu \left(x\right)Q\left(x\right)$$\mu (x)\frac{dy}{dx}+\mu (x)P(x)y=\mu (x)Q(x)$

5. The above result is equivalent to, by Chain Rule for derivatives: $\frac{d}{dx}\left[\mu \left(x\right)y\right]=\mu \left(x\right)Q\left(x\right)$$\frac{d}{dx}\left [\mu (x)y\right]=\mu (x)Q(x)$

6. Integrate both sides of the DE: $\int \frac{d}{dx}\left[\mu \left(x\right)y\right]=\int \mu \left(x\right)Q\left(x\right)dx$$\int\frac{d}{dx}\left [\mu (x)y\right]=\int\mu (x)Q(x)dx$

7. The above is equivalent to, by FTC-1: $\mu \left(x\right)y=\int \mu \left(x\right)Q\left(x\right)dx$$\mu (x)y=\int\mu (x)Q(x)dx$

8. Divide both sides of above equation: $y=\frac{1}{\mu \left(x\right)}\int \mu \left(x\right)Q\left(x\right)dx$$y=\frac{1}{\mu (x)}\int\mu (x)Q(x)dx$

9. Solve for $y$$y$, i.e., find an explicit solution, if possible. If it is not possible to solve for $y$$y$, leave the answer as an implicit solution;

10. If there is an initial condition, $y\left({x}_{0}\right)={y}_{0}$$\displaystyle y(x_0)=y_0$, use it to find the constant, $C$$C$;

11. Check for any missing solutions.

NOTE: If the solution to an ODE contains an integral that does not have an analytical antiderivative, then write the solution as an integral equation.