Solve dy/dx+p(x)y=dx

$$\frac { d y } { d x } + p ( x ) y = d x$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$dx=\frac{\mathrm{d}(y)}{\mathrm{d}x}+pxy$$

The equation is in standard form.

$$xd=pxy$$

Divide both sides by x.

$$\frac{xd}{x}=\frac{pxy}{x}$$

Dividing by x undoes the multiplication by x.

$$d=\frac{pxy}{x}$$

Divide pxy by x.

$$d=py$$

Solve for p (complex solution)

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

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(y)}{\mathrm{d}x} from both sides.

$$pxy=dx-\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

The equation is in standard form.

$$xyp=dx$$

Divide both sides by xy.

$$\frac{xyp}{xy}=\frac{dx}{xy}$$

Dividing by xy undoes the multiplication by xy.

$$p=\frac{dx}{xy}$$

Divide dx by xy.

$$p=\frac{d}{y}$$

Solve for d

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

Steps for Solving Linear Equation

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

$$dx=\frac{\mathrm{d}(y)}{\mathrm{d}x}+pxy$$

The equation is in standard form.

$$xd=pxy$$

Divide both sides by x.

$$\frac{xd}{x}=\frac{pxy}{x}$$

Dividing by x undoes the multiplication by x.

$$d=\frac{pxy}{x}$$

Divide pxy by x.

$$d=py$$

Solve for p

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

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}(y)}{\mathrm{d}x} from both sides.

$$pxy=dx-\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

The equation is in standard form.

$$xyp=dx$$

Divide both sides by xy.

$$\frac{xyp}{xy}=\frac{dx}{xy}$$

Dividing by xy undoes the multiplication by xy.

$$p=\frac{dx}{xy}$$

Divide dx by xy.

$$p=\frac{d}{y}$$