How Do You Solve a Differential Equation Using an Integrating Factor?

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First identify $P(x)$ and $Q(x)$. The integrating factor is

Do not include an arbitrary constant in this integral: any non-zero constant multiple produces an equivalent integrating factor.

Multiply the entire equation by $\mu(x)$, not only the terms involving $y$. Because $\mu'(x)=P(x)\mu(x)$, the product rule gives

The key mechanism is the product rule. Multiplication by $\mu(x)$ creates the terms $\mu y'$ and $\mu' y$. Since $\mu'=P\mu$, these match $\mu y'+P\mu y$, so the left-hand side is exactly $(\mu y)'$. This explains why the method works.

Integrate both sides and divide by $\mu(x)$:

Worked Example

Solve

Here $P(x)=2$, so

Multiplying every term by $e^{2x}$ gives

Therefore,

so

and hence

For example, if $y(0)=4$, then $\frac{1}{3}+C=4$, giving $C=\frac{11}{3}$. The particular solution is

A useful verification is to differentiate the final expression and substitute it into the original equation. In the general solution, the derivative contains one term from the particular solution and one from the complementary solution. The complementary terms cancel when you form $y'+2y$, while the remaining terms simplify to $e^x$. This confirms both the algebra and the sign of the exponent. It is also a quick way to detect a missing constant.

Exam Technique

Show the standard form, integrating factor, multiplication of every term, and product derivative explicitly. Substitute any initial condition only after finding the general solution.

A common misconception is treating the coefficient of (y') as $P(x)$. First divide the whole equation by that coefficient so the coefficient of (y') is $1$; otherwise, the integrating factor is incorrect.