Solve operatorname{ve}d^2y/dx^2-y=e^3xcos2x-e^2xsin3

$$\operatorname { ve } \frac { d ^ { 2 } y } { d x ^ { 2 } } - y = e ^ { 3 x } \cos 2 x - e ^ { 2 x } \sin 3$$

Solve for v (complex solution)

$$v\in \mathrm{C}$$

$$y=-\frac{e^{2x}\left(e^{\left(1-2i\right)x}+e^{\left(1+2i\right)x}-2\cos(\frac{\pi }{2}-3)\right)}{2}$$

Steps for Solving Linear Equation

Add y to both sides.

$$ve\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}=e^{3x}\cos(2x)-e^{2x}\sin(3)+y$$

The equation is in standard form.

$$0=\cos(2x)e^{3x}-\sin(3)e^{2x}+y$$

This is false for any v.

$$v\in$$

Solve for v

$$v\in \mathrm{R}$$

$$e^{2x}\left(\cos(2x)e^{x}-\sin(3)\right)+y=0$$

Steps for Solving Linear Equation

Add y to both sides.

$$ve\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}=e^{3x}\cos(2x)-e^{2x}\sin(3)+y$$

The equation is in standard form.

$$0=\cos(2x)e^{3x}-\sin(3)e^{2x}+y$$

This is false for any v.

$$v\in$$