Integrating Factor

When solving first order linear differential equations of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'\mu (x) + p(x)\mu (x)y } , can now be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mu (x)y)' } since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'\mu (x) + p(x)\mu (x)y = y'\mu (x) + \mu '(x)y} . Verify by taking the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu (x)y } with respect to x with the product rule.
4. Now, integrate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mu (x)y)' = g(x)\mu (x)} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu (x)y = \int g(x)\mu (x)dx } .
5. Solve for y.

Resources

• Solving Linear Equations, Paul's Online Notes