Solve xdy/dx+y=1/y^2

$$\frac { x d y } { d x } + y = \frac { 1 } { y ^ { 2 } }$$

Solve for d

$$d\neq 0$$

$$x\neq 0\text{ and }y=\frac{2^{\frac{2}{3}}}{2}$$

Steps for Solving Linear Equation

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dxy^{2}, the least common multiple of dx,y^{2}.

$$y^{2}xdy+dxy^{2}y=xd$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}xd+dxy^{2}y=xd$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}xd+dxy^{3}=xd$$

Combine y^{3}xd and dxy^{3} to get 2y^{3}xd.

$$2y^{3}xd=xd$$

Subtract xd from both sides.

$$2y^{3}xd-xd=0$$

Combine all terms containing d.

$$\left(2y^{3}x-x\right)d=0$$

The equation is in standard form.

$$\left(2xy^{3}-x\right)d=0$$

Divide 0 by 2y^{3}x-x.

$$d=0$$

Variable d cannot be equal to 0.

$$d\in \emptyset$$

Solve for x

$$x\neq 0$$

$$d\neq 0\text{ and }y=\frac{2^{\frac{2}{3}}}{2}$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dxy^{2}, the least common multiple of dx,y^{2}.

$$y^{2}xdy+dxy^{2}y=dx$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}xd+dxy^{2}y=dx$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}xd+dxy^{3}=dx$$

Combine y^{3}xd and dxy^{3} to get 2y^{3}xd.

$$2y^{3}xd=dx$$

Subtract dx from both sides.

$$2y^{3}xd-dx=0$$

Combine all terms containing x.

$$\left(2y^{3}d-d\right)x=0$$

The equation is in standard form.

$$\left(2dy^{3}-d\right)x=0$$

Divide 0 by 2y^{3}d-d.

$$x=0$$

Variable x cannot be equal to 0.

$$x\in \emptyset$$