Solve 3dy+(3y-e^2x)dx=0

$$3 d y + ( 3 y - e ^ { 2 x } ) d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=\frac{xe^{2x}}{3\left(x+1\right)}\text{ and }x\neq -1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3y-e^{2x} by d.

$$3dy+\left(3yd-e^{2x}d\right)x=0$$

Use the distributive property to multiply 3yd-e^{2x}d by x.

$$3dy+3ydx-e^{2x}dx=0$$

Combine all terms containing d.

$$\left(3y+3yx-e^{2x}x\right)d=0$$

The equation is in standard form.

$$\left(3xy-xe^{2x}+3y\right)d=0$$

Divide 0 by 3y+3yx-e^{2x}x.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=\frac{xe^{2x}}{3\left(x+1\right)}\text{ and }x\neq -1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3y-e^{2x} by d.

$$3dy+\left(3yd-e^{2x}d\right)x=0$$

Use the distributive property to multiply 3yd-e^{2x}d by x.

$$3dy+3ydx-e^{2x}dx=0$$

Combine all terms containing d.

$$\left(3y+3yx-e^{2x}x\right)d=0$$

The equation is in standard form.

$$\left(3xy-xe^{2x}+3y\right)d=0$$

Divide 0 by 3y+3yx-e^{2x}x.

$$d=0$$