Solve dy/dx=y(t)+y(x-1)

$$\frac { d y } { d x } = y ( t ) + y ( x - 1 )$$

Solve for t

$$\left\{\begin{matrix}\\t=1-x\text{, }&\text{unconditionally}\\t\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y by x-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=yt+yx-y$$

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

$$yt+yx-y=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract yx from both sides.

$$yt-y=\frac{\mathrm{d}}{\mathrm{d}x}(y)-yx$$

Add y to both sides.

$$yt=\frac{\mathrm{d}}{\mathrm{d}x}(y)-yx+y$$

The equation is in standard form.

$$yt=y-xy$$

Divide both sides by y.

$$\frac{yt}{y}=\frac{y-xy}{y}$$

Dividing by y undoes the multiplication by y.

$$t=\frac{y-xy}{y}$$

Divide -yx+y by y.

$$t=1-x$$

Solve for x

$$\left\{\begin{matrix}\\x=1-t\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$