Solve (2xcosy+3x^2y)dx+(x^3-x^2siny-y)dy=0

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

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y\left(-x^{2}\sin(y)+4x^{3}-y\right)+2x^{2}\cos(y)=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

Use the distributive property to multiply \left(x^{3}-x^{2}\sin(y)\right)d-yd by y.

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

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

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

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

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

Combine 3ydx^{3} and x^{3}dy to get 4ydx^{3}.

$$2\cos(y)dx^{2}+4ydx^{3}-x^{2}\sin(y)dy-dy^{2}=0$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide 0 by 2\cos(y)x^{2}+4yx^{3}-x^{2}\sin(y)y-y^{2}.

$$d=0$$