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

3. Compute the integrating factor: ${\displaystyle \mu (x)=e^{\int P(x)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 (x)$$\mu (x)$ to obtain $\mu (x)\frac{dy}{dx}+\mu (x)P(x)y=\mu (x)Q(x)$$\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}[\mu (x)y]=\mu (x)Q(x)$$\frac{d}{dx}\left [\mu (x)y\right]=\mu (x)Q(x)$

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

8. Divide both sides of above equation: $y=\frac{1}{\mu (x)}\int \mu (x)Q(x)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, ${\displaystyle y(x_0)=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.

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Last Modified: Wednesday, 2 September 2020 07:22 EDT