Solve y^prime(t)=x(t)+py(t)

$$y ^ { \prime } ( t ) = x ( t ) + p y ( t )$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{x}{y}\text{, }&y\neq 0\\p\in \mathrm{R}\text{, }&t=0\text{ or }\left(x=0\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$xt+pyt=\frac{\mathrm{d}}{\mathrm{d}x}(y)t$$

Subtract xt from both sides.

$$pyt=\frac{\mathrm{d}}{\mathrm{d}x}(y)t-xt$$

The equation is in standard form.

$$typ=-tx$$

Divide both sides by yt.

$$\frac{typ}{ty}=-\frac{tx}{ty}$$

Dividing by yt undoes the multiplication by yt.

$$p=-\frac{tx}{ty}$$

Divide -xt by yt.

$$p=-\frac{x}{y}$$

Solve for t

$$\left\{\begin{matrix}\\t=0\text{, }&\text{unconditionally}\\t\in \mathrm{R}\text{, }&x=-py\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract xt from both sides.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)t-xt=pyt$$

Subtract pyt from both sides.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)t-xt-pyt=0$$

Reorder the terms.

$$t\frac{\mathrm{d}}{\mathrm{d}x}(y)-pty-tx=0$$

Combine all terms containing t.

$$\left(\frac{\mathrm{d}}{\mathrm{d}x}(y)-py-x\right)t=0$$

The equation is in standard form.

$$\left(-x-py\right)t=0$$

Divide 0 by -py-x.

$$t=0$$