Integrating Factor

When solving first order linear differential equations of the form $y'+p(x)y=g(x)$, we can utilize the "integrating factor" $\mu (x)=e^{\int p(x)dx}$.

Steps to solving an equation of the form \frac{dy}{dx} + p(x)y = g(x):

1. Find the integrating factor $\mu (x)=e^{\int p(x)dx}$, and note that $\mu '(x)=p(x)e^{\int p(x)dx}=p(x)\mu (x)$,
2. Multiply both sides of the equation by the integrating factor.
3. The left side of the equation, $y'\mu (x)+p(x)\mu (x)y$, can now be rewritten as $(\mu (x)y)'$ since $y'\mu (x)+p(x)\mu (x)y=y'\mu (x)+\mu '(x)y$. Verify by taking the derivative of $\mu (x)y$ with respect to x with the product rule.
4. Now, integrate $(\mu (x)y)'=g(x)\mu (x)$ to get $\mu (x)y=\int g(x)\mu (x)dx$.
5. Solve for y.

Resources

• Solving Linear Equations, Paul's Online Notes