Solve (y+y^3/3+x^2/2)dx+1/4(x+xy^2)dy=0

$$( y + \frac { y ^ { 3 } } { 3 } + \frac { x ^ { 2 } } { 2 } ) d x + \frac { 1 } { 4 } ( x + x y ^ { 2 } ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }y=\frac{7^{\frac{2}{3}}}{7}\left(\sqrt[3]{\frac{\sqrt{441x^{4}+875}}{7}-3x^{2}}-\sqrt[3]{\frac{\sqrt{441x^{4}+875}}{7}+3x^{2}}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$12\left(y+\frac{y^{3}}{3}+\frac{x^{2}}{2}\right)dx+3\left(x+xy^{2}\right)dy=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{y^{3}}{3} times \frac{2}{2}. Multiply \frac{x^{2}}{2} times \frac{3}{3}.

$$12\left(y+\frac{2y^{3}}{6}+\frac{3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

Since \frac{2y^{3}}{6} and \frac{3x^{2}}{6} have the same denominator, add them by adding their numerators.

$$12\left(y+\frac{2y^{3}+3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

Use the distributive property to multiply 12 by y+\frac{2y^{3}+3x^{2}}{6}.

$$\left(12y+12\times \frac{2y^{3}+3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

Cancel out 6, the greatest common factor in 12 and 6.

$$\left(12y+2\left(2y^{3}+3x^{2}\right)\right)dx+3\left(x+xy^{2}\right)dy=0$$

Use the distributive property to multiply 2 by 2y^{3}+3x^{2}.

$$\left(12y+4y^{3}+6x^{2}\right)dx+3\left(x+xy^{2}\right)dy=0$$

Use the distributive property to multiply 12y+4y^{3}+6x^{2} by d.

$$\left(12yd+4y^{3}d+6x^{2}d\right)x+3\left(x+xy^{2}\right)dy=0$$

Use the distributive property to multiply 12yd+4y^{3}d+6x^{2}d by x.

$$12ydx+4y^{3}dx+6dx^{3}+3\left(x+xy^{2}\right)dy=0$$

Use the distributive property to multiply 3 by x+xy^{2}.

$$12ydx+4y^{3}dx+6dx^{3}+\left(3x+3xy^{2}\right)dy=0$$

Use the distributive property to multiply 3x+3xy^{2} by d.

$$12ydx+4y^{3}dx+6dx^{3}+\left(3xd+3xy^{2}d\right)y=0$$

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

$$12ydx+4y^{3}dx+6dx^{3}+3xdy+3xdy^{3}=0$$

Combine 12ydx and 3xdy to get 15ydx.

$$15ydx+4y^{3}dx+6dx^{3}+3xdy^{3}=0$$

Combine 4y^{3}dx and 3xdy^{3} to get 7y^{3}dx.

$$15ydx+7y^{3}dx+6dx^{3}=0$$

Combine all terms containing d.

$$\left(15yx+7y^{3}x+6x^{3}\right)d=0$$

The equation is in standard form.

$$\left(6x^{3}+7xy^{3}+15xy\right)d=0$$

Divide 0 by 15yx+7y^{3}x+6x^{3}.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{-42y^{3}-90y}}{6}\text{; }x=-\frac{\sqrt{-42y^{3}-90y}}{6}\text{, }&y\leq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$