Solve (5x^2+siny)dx+(2x^3+2xsiny+2xcosy)dy=0

$$( 5 x ^ { 2 } + \sin y ) d x + ( 2 x ^ { 3 } + 2 x \sin y + 2 x \cos y ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }\left(y\neq -\frac{5}{2}\text{ and }\frac{2y\sin(y)+2y\cos(y)+\sin(y)}{2y+5}\leq 0\text{ and }|x|=\sqrt{-\frac{2y\sin(y)+2y\cos(y)+\sin(y)}{2y+5}}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 5x^{2}+\sin(y) by d.

$$\left(5x^{2}d+\sin(y)d\right)x+\left(2x^{3}+2x\sin(y)+2x\cos(y)\right)dy=0$$

Use the distributive property to multiply 5x^{2}d+\sin(y)d by x.

$$5dx^{3}+\sin(y)dx+\left(2x^{3}+2x\sin(y)+2x\cos(y)\right)dy=0$$

Use the distributive property to multiply 2x^{3}+2x\sin(y)+2x\cos(y) by d.

$$5dx^{3}+\sin(y)dx+\left(2x^{3}d+2x\sin(y)d+2x\cos(y)d\right)y=0$$

Use the distributive property to multiply 2x^{3}d+2x\sin(y)d+2x\cos(y)d by y.

$$5dx^{3}+\sin(y)dx+2x^{3}dy+2x\sin(y)dy+2x\cos(y)dy=0$$

Combine all terms containing d.

$$\left(5x^{3}+\sin(y)x+2x^{3}y+2x\sin(y)y+2x\cos(y)y\right)d=0$$

The equation is in standard form.

$$\left(2xy\sin(y)+2xy\cos(y)+x\sin(y)+2yx^{3}+5x^{3}\right)d=0$$

Divide 0 by 5x^{3}+\sin(y)x+2x^{3}y+2x\sin(y)y+2x\cos(y)y.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\sqrt{-\frac{2y\sin(y)+2y\cos(y)+\sin(y)}{2y+5}}\text{; }x=-\sqrt{-\frac{2y\sin(y)+2y\cos(y)+\sin(y)}{2y+5}}\text{, }&y\neq -\frac{5}{2}\text{ and }\frac{2y\sin(y)+2y\cos(y)+\sin(y)}{2y+5}\leq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$