( m ^ { 5 } + 3 y ) d x - n d y = 0
Solve for d
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(n=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=0\right)\text{ or }\left(m=\frac{\sqrt[5]{y\left(n-3x\right)}}{\sqrt[5]{x}}\text{ and }x\neq 0\right)\end{matrix}\right.
Solve for m
\left\{\begin{matrix}m=\sqrt[5]{\frac{y\left(n-3x\right)}{x}}\text{, }&x\neq 0\\m\in \mathrm{R}\text{, }&d=0\text{ or }\left(n=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=0\right)\end{matrix}\right.
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\left(m^{5}d+3yd\right)x-ndy=0
Use the distributive property to multiply m^{5}+3y by d.
m^{5}dx+3ydx-ndy=0
Use the distributive property to multiply m^{5}d+3yd by x.
3dxy+dxm^{5}-dny=0
Reorder the terms.
\left(3xy+xm^{5}-ny\right)d=0
Combine all terms containing d.
d=0
Divide 0 by 3xy+xm^{5}-ny.
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