Solve (2x(y)^(4)(e)^(y)+2x(y)^(3)+y)dx+((x)^(2)(y)^(4)(e)^(y)-(x)^(2)(y)^(2)-3x)dy=0 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(2x { y }^{ 4 } { e }^{ y } +2x { y }^{ 3 } +y)dx+( { x }^{ 2 } { y }^{ 4 } { e }^{ y } - { x }^{ 2 } { y }^{ 2 } -3x)dy=0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply $2xy^{4}e^{y}+2xy^{3}+y$ by $d$.

$$\left(2xy^{4}e^{y}d+2xy^{3}d+yd\right)x+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x\right)dy=0$$

Use the distributive property to multiply $2xy^{4}e^{y}d+2xy^{3}d+yd$ by $x$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x\right)dy=0$$

Use the distributive property to multiply $x^{2}y^{4}e^{y}-x^{2}y^{2}-3x$ by $d$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)d-3xd\right)y=0$$

Use the distributive property to multiply $\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)d-3xd$ by $y$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)dy-3xdy=0$$

Combine $ydx$ and $-3xdy$ to get $-2ydx$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)dy=0$$

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

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+\left(x^{2}y^{4}e^{y}d-x^{2}y^{2}d\right)y=0$$

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

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+x^{2}e^{y}dy^{5}-x^{2}dy^{3}=0$$

Combine $2y^{3}dx^{2}$ and $-x^{2}dy^{3}$ to get $y^{3}dx^{2}$.

$$2y^{4}e^{y}dx^{2}+y^{3}dx^{2}-2ydx+x^{2}e^{y}dy^{5}=0$$

Combine all terms containing $d$.

$$\left(2y^{4}e^{y}x^{2}+y^{3}x^{2}-2yx+x^{2}e^{y}y^{5}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{5}e^{y}+2x^{2}y^{4}e^{y}+x^{2}y^{3}-2xy\right)d=0$$

Divide $0$ by $2y^{4}e^{y}x^{2}+y^{3}x^{2}-2yx+x^{2}e^{y}y^{5}$.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply $2xy^{4}e^{y}+2xy^{3}+y$ by $d$.

$$\left(2xy^{4}e^{y}d+2xy^{3}d+yd\right)x+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x\right)dy=0$$

Use the distributive property to multiply $2xy^{4}e^{y}d+2xy^{3}d+yd$ by $x$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x\right)dy=0$$

Use the distributive property to multiply $x^{2}y^{4}e^{y}-x^{2}y^{2}-3x$ by $d$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)d-3xd\right)y=0$$

Use the distributive property to multiply $\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)d-3xd$ by $y$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}+ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)dy-3xdy=0$$

Combine $ydx$ and $-3xdy$ to get $-2ydx$.

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+\left(x^{2}y^{4}e^{y}-x^{2}y^{2}\right)dy=0$$

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

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+\left(x^{2}y^{4}e^{y}d-x^{2}y^{2}d\right)y=0$$

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

$$2y^{4}e^{y}dx^{2}+2y^{3}dx^{2}-2ydx+x^{2}e^{y}dy^{5}-x^{2}dy^{3}=0$$

Combine $2y^{3}dx^{2}$ and $-x^{2}dy^{3}$ to get $y^{3}dx^{2}$.

$$2y^{4}e^{y}dx^{2}+y^{3}dx^{2}-2ydx+x^{2}e^{y}dy^{5}=0$$

Combine all terms containing $d$.

$$\left(2y^{4}e^{y}x^{2}+y^{3}x^{2}-2yx+x^{2}e^{y}y^{5}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{5}e^{y}+2x^{2}y^{4}e^{y}+x^{2}y^{3}-2xy\right)d=0$$

Divide $0$ by $2y^{4}e^{y}x^{2}+y^{3}x^{2}-2yx+x^{2}e^{y}y^{5}$.

$$d=0$$

Solve for x (complex solution)

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

Solve for x

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