I'm taking a college course in differential equations after a long break from math classes, and I'm struggling with finding general solutions through integrating factors. Here is the "easy" question from the recent homework (which I got wrong).
Find the general solution of the given differential equation: $$4\frac{dy}{dx} + 20y = 5$$
I know that $\frac{dy}{dx}$ needs a coefficient of one, so I rewrote the equation as $\frac{dy}{dx} + 4y = 5/4$. I found the integrating factor to be $e^{5x}$. Attached is the rest of my work for this problem. Where am I going wrong?
Thank you.
After you obtain the integrating factor $I$, you multiply the $\textbf{full}$ ODE with $I$, which includes the RHS. So multiplying through by $e^{5x}$ we obtain
$$\frac{dy}{dx}+5y=\frac{5}{4}$$ $$e^{5x}\frac{dy}{dx}+5e^{5x}y=\frac{5}{4}\color{red}{e^{5x}}$$ $$\frac{d}{dx}\left(e^{5x}y\right)=\frac{5}{4}e^{5x}$$ $$e^{5x}y=\frac{1}{4}e^{5x}+C$$ $$y=\frac{1}{4}+Ce^{-5x}$$