Solve e^4x*ydy=(y+5)dx

$$e ^ { 4 x } \cdot y d y = ( y + 5 ) d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=\frac{\sqrt{x^{2}+20xe^{4x}}+x}{2e^{4x}}\text{ or }y=\frac{-\sqrt{x^{2}+20xe^{4x}}+x}{2e^{4x}}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$e^{4x}y^{2}d=\left(y+5\right)dx$$

Use the distributive property to multiply y+5 by d.

$$e^{4x}y^{2}d=\left(yd+5d\right)x$$

Use the distributive property to multiply yd+5d by x.

$$e^{4x}y^{2}d=ydx+5dx$$

Subtract ydx from both sides.

$$e^{4x}y^{2}d-ydx=5dx$$

Subtract 5dx from both sides.

$$e^{4x}y^{2}d-ydx-5dx=0$$

Reorder the terms.

$$-dxy-5dx+dy^{2}e^{4x}=0$$

Combine all terms containing d.

$$\left(-xy-5x+y^{2}e^{4x}\right)d=0$$

The equation is in standard form.

$$\left(y^{2}e^{4x}-5x-xy\right)d=0$$

Divide 0 by -xy-5x+y^{2}e^{4x}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(y=\frac{\sqrt{x\left(x+20e^{4x}\right)}+x}{2e^{4x}}\text{ or }y=\frac{-\sqrt{x\left(x+20e^{4x}\right)}+x}{2e^{4x}}\right)\text{ and }x\left(x+20e^{4x}\right)\geq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$e^{4x}y^{2}d=\left(y+5\right)dx$$

Use the distributive property to multiply y+5 by d.

$$e^{4x}y^{2}d=\left(yd+5d\right)x$$

Use the distributive property to multiply yd+5d by x.

$$e^{4x}y^{2}d=ydx+5dx$$

Subtract ydx from both sides.

$$e^{4x}y^{2}d-ydx=5dx$$

Subtract 5dx from both sides.

$$e^{4x}y^{2}d-ydx-5dx=0$$

Reorder the terms.

$$-dxy-5dx+dy^{2}e^{4x}=0$$

Combine all terms containing d.

$$\left(-xy-5x+y^{2}e^{4x}\right)d=0$$

The equation is in standard form.

$$\left(y^{2}e^{4x}-5x-xy\right)d=0$$

Divide 0 by -xy-5x+y^{2}e^{4x}.

$$d=0$$