Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 504: 8

See the details below.

Since $ y=e^x \sin 2x $, then $$ y'=e^x\sin 2x+2e^x\cos 2x=e^x(\sin 2x+2\cos 2x), \\ y'' =e^x(\sin 2x+2\cos 2x)+e^x(2\cos 2x-4\sin 2x).$$ Now, by substitution of $ y $, $ y'$ and $ y''$ into the equation, we verify that $ y $ is a solution of the given differential equation.