Solve dy/dx+P(y)=Q(x)

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

Solve for P

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

Steps for Solving Linear Equation

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

$$Py=Qx-\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

The equation is in standard form.

$$yP=Qx$$

Divide both sides by y.

$$\frac{yP}{y}=\frac{Qx}{y}$$

Dividing by y undoes the multiplication by y.

$$P=\frac{Qx}{y}$$

Solve for Q

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

Steps for Solving Linear Equation

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

$$Qx=\frac{\mathrm{d}}{\mathrm{d}x}(y)+Py$$

The equation is in standard form.

$$xQ=Py$$

Divide both sides by x.

$$\frac{xQ}{x}=\frac{Py}{x}$$

Dividing by x undoes the multiplication by x.

$$Q=\frac{Py}{x}$$