Solve y(4+e^x)dy-e^xdx=0

$$y ( 4 + e ^ { x } ) d y - e ^ { x } d x = 0$$

Solve for d

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

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply y^{2} by 4+e^{x}.

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

Use the distributive property to multiply 4y^{2}+y^{2}e^{x} by d.

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

Reorder the terms.

$$-dxe^{x}+dy^{2}e^{x}+4dy^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$