Solve the following differential equation by undetermined coefficients.
$$y''+4y=3\sin(2x)$$
Characteristic polynomial:
$$r^2+4=0, e=\pm2i$$
$$y_c=c_1\cos(2x)+c_2\sin(2x)$$
$$y_p=Ax\cos(2x)+Bx\sin(2x)$$
$$y'_p=A\cos(2x)-2Ax\sin(2x)+B\sin(2x)+2Bx\cos(2x)$$
$$y''_p=-2A\sin(2x)-2A\sin(2x)-4x\cos(2x)-2B\cos(2x)+2B\cos(2x)-$$
$$4Bx\sin(2x)$$
$$=-4A\sin(2x)-4x\cos(2x)-4B\sin(2x)=3x\sin$$
$$A=0,B=-\frac{3}{4}, y=c_1\cos(2x)+c_2\sin(2x)-\frac{3}{4}x\sin(2x)$$
Answer in the book is$y=c_1\cos(2x)+c_2\sin(2x)-\frac{3}{4}x\cos(2x)$

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