## Calculus 8th Edition

$2e^x=2e^x$
given: $y=\frac{2}{3}e^x+e^{-2x}$ $y'=\frac{dy}{dx}$ $=\frac{2}{3}e^x-2e^{-2x}$ now we have expressions for both y and y', which we can plug into our differential equation. If this produces a true statement, then the given expression for y is a valid solution. $y'+2y=2e^x$ $\frac{2}{3}e^x-2e^{-2x}+2[\frac{2}{3}e^x+e^{-2x}]=2e^x$ $\frac{2}{3}e^x-2e^{-2x}+\frac{4}{3}e^x+2e^{-2x}=2e^x$ $2e^x=2e^x$