Solve f(x)dy+2yf^prime(x)dx=f^(x)*f^prime(x)dx

$$f ( x ) d y + 2 y f ^ { \prime } ( x ) d x = f ^ { ( x ) } \cdot f ^ { \prime } ( x ) d x$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)xdx$$

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

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d$$

Subtract f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d from both sides.

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d-f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=0$$

Reorder the terms.

$$-dx^{2}f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)+2dyx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(f)+dfxy=0$$

Combine all terms containing d.

$$\left(-x^{2}f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)+2yx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(f)+fxy\right)d=0$$

The equation is in standard form.

$$fxyd=0$$

Divide 0 by fxy.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&\left(f<0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(f=0\text{ and }x>0\right)\text{ or }f>0\\d\in \mathrm{R}\text{, }&\left(x=0\text{ and }f\neq 0\right)\text{ or }\left(y=0\text{ and }f<0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(f=0\text{ and }x>0\right)\text{ or }\left(y=0\text{ and }f>0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)xdx$$

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

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d$$

Subtract f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d from both sides.

$$fxdy+2y\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d-f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)x^{2}d=0$$

Reorder the terms.

$$-dx^{2}f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)+2dyx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(f)+dfxy=0$$

Combine all terms containing d.

$$\left(-x^{2}f^{x}\frac{\mathrm{d}}{\mathrm{d}x}(f)+2yx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(f)+fxy\right)d=0$$

The equation is in standard form.

$$fxyd=0$$

Divide 0 by fxy.

$$d=0$$

Solve for f (complex solution)

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

Solve for f

$$\left\{\begin{matrix}f=0\text{, }&x>0\\f<0\text{, }&x=0\text{ or }\left(d=0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(y=0\text{ and }Denominator(x)\text{bmod}2=1\right)\\f>0\text{, }&y=0\text{ or }d=0\text{ or }x=0\end{matrix}\right.$$