Solve d^2x/dx^2+dx/dt=acosx

$$\frac { d ^ { 2 } x } { d x ^ { 2 } } + \frac { d x } { d t } = a \cos x$$

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$a\cos(x)=\frac{\mathrm{d}(x)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(x)}{\mathrm{d}t}$$

The equation is in standard form.

$$\cos(x)a=0$$

Divide 0 by \cos(x).

$$a=0$$

Solve for t

$$t\in \mathrm{R}$$

$$\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\text{ or }a=0$$