First-Order Linear EquationsExpected Educational ResultsLinear EquationsMethod for Solving First-Order ODEsCC BY-NC-SA 4.0

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

**Section 2.6**– Linear Equations

**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.

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

Rewrite the DE in standard form,

;$\frac{dy}{dx}+P(x)y=Q(x)$ Compute the integrating factor:

[no$\mu (x)={e}^{\int P(x)dx}$ needed here];$+C$ Multiply both sides of the DE in standard form by:

to obtain$\mu (x)$ $\mu (x)\frac{dy}{dx}+\mu (x)P(x)y=\mu (x)Q(x)$ The above result is equivalent to, by Chain Rule for derivatives:

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

$\int \frac{d}{dx}[\mu (x)y]=\int \mu (x)Q(x)dx$ The above is equivalent to, by FTC-1:

$\mu (x)y=\int \mu (x)Q(x)dx$ Divide both sides of above equation:

$y=\frac{1}{\mu (x)}\int \mu (x)Q(x)dx$ Solve for

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

, use it to find the constant,$y({x}_{0})={y}_{0}$ ;$C$ 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