Solve taudY(t)/dt+Y(t)=k_{p}U(t)

$$\tau \frac { d Y ( t ) } { d t } + Y ( t ) = k _ { p } U ( t )$$

Solve for U

$$\left\{\begin{matrix}U=\frac{Y\left(t+2\pi \right)}{k_{p}t}\text{, }&t\neq 0\text{ and }k_{p}\neq 0\\U\in \mathrm{R}\text{, }&\left(Y=0\text{ and }k_{p}=0\right)\text{ or }\left(Y=0\text{ and }t=0\right)\text{ or }\left(t=-2\pi \text{ and }k_{p}=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Substitute 2 * \pi for \tau.

$$2 * \pi \frac{d}{dt} {Y {(t)}} + Y {(t)} = k_{p} U {(t)}$$

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

$$k_{p}Ut=2\pi \frac{\mathrm{d}}{\mathrm{d}t}(Yt)+Yt$$

The equation is in standard form.

$$k_{p}tU=Yt+2\pi Y$$

Divide both sides by k_{p}t.

$$\frac{k_{p}tU}{k_{p}t}=\frac{Y\left(t+2\pi \right)}{k_{p}t}$$

Dividing by k_{p}t undoes the multiplication by k_{p}t.

$$U=\frac{Y\left(t+2\pi \right)}{k_{p}t}$$

Solve for Y

$$\left\{\begin{matrix}Y=\frac{Uk_{p}t}{t+2\pi }\text{, }&t\neq -2\pi \\Y\in \mathrm{R}\text{, }&\left(U=0\text{ or }k_{p}=0\right)\text{ and }t=-2\pi \end{matrix}\right.$$