Solve xdy-2xydx=(x-2)e^ x dx | 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

$$xdy-2xydx=(x-2)e^{x}dx$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$xdy-2x^{2}yd=\left(x-2\right)e^{x}dx$$

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

$$xdy-2x^{2}yd=\left(xe^{x}-2e^{x}\right)dx$$

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

$$xdy-2x^{2}yd=\left(xe^{x}d-2e^{x}d\right)x$$

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

$$xdy-2x^{2}yd=e^{x}dx^{2}-2e^{x}dx$$

Subtract $e^{x}dx^{2}$ from both sides.

$$xdy-2x^{2}yd-e^{x}dx^{2}=-2e^{x}dx$$

Add $2e^{x}dx$ to both sides.

$$xdy-2x^{2}yd-e^{x}dx^{2}+2e^{x}dx=0$$

Reorder the terms.

$$-dx^{2}e^{x}+dxy+2dxe^{x}-2dyx^{2}=0$$

Combine all terms containing $d$.

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

The equation is in standard form.

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

Divide $0$ by $-x^{2}e^{x}+xy+2xe^{x}-2yx^{2}$.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

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

$$xdy-2x^{2}yd=\left(x-2\right)e^{x}dx$$

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

$$xdy-2x^{2}yd=\left(xe^{x}-2e^{x}\right)dx$$

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

$$xdy-2x^{2}yd=\left(xe^{x}d-2e^{x}d\right)x$$

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

$$xdy-2x^{2}yd=e^{x}dx^{2}-2e^{x}dx$$

Subtract $e^{x}dx^{2}$ from both sides.

$$xdy-2x^{2}yd-e^{x}dx^{2}=-2e^{x}dx$$

Add $2e^{x}dx$ to both sides.

$$xdy-2x^{2}yd-e^{x}dx^{2}+2e^{x}dx=0$$

Reorder the terms.

$$-dx^{2}e^{x}+dxy+2dxe^{x}-2dyx^{2}=0$$

Combine all terms containing $d$.

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

The equation is in standard form.

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

Divide $0$ by $-x^{2}e^{x}+xy+2xe^{x}-2yx^{2}$.

$$d=0$$