Calculus Examples

$x^{2} + x y + y^{2} - y^{2} \frac{d x}{d y} = 0$

Step 1

Rewrite the differential equation as a function of $\frac{x}{y}$ .

Tap for more steps...

Step 1.1

Solve for $\frac{d x}{d y}$ .

Tap for more steps...

Step 1.1.1

Move all terms not containing $\frac{d x}{d y}$ to the right side of the equation.

Tap for more steps...

Step 1.1.1.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 1.1.1.2

Subtract $x y$ from both sides of the equation.

$y^{2} - y^{2} \frac{d x}{d y} = - x^{2} - x y$

Step 1.1.1.3

Subtract $y^{2}$ from both sides of the equation.

$- y^{2} \frac{d x}{d y} = - x^{2} - x y - y^{2}$

Step 1.1.2

Divide each term in $- y^{2} \frac{d x}{d y} = - x^{2} - x y - y^{2}$ by $- y^{2}$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $- y^{2} \frac{d x}{d y} = - x^{2} - x y - y^{2}$ by $- y^{2}$ .

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

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.2.2

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

Divide $\frac{d x}{d y}$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.3

Cancel the common factor of $y$ and $y^{2}$ .

Tap for more steps...

Step 1.1.2.3.1.3.1

Factor $y$ out of $x y$ .

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

Step 1.1.2.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.3.1.3.2.1

Factor $y$ out of $y^{2}$ .

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

Step 1.1.2.3.1.3.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.3.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.4

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.5

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 1.1.2.3.1.5.1

Cancel the common factor.

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

Step 1.1.2.3.1.5.2

Rewrite the expression.

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

Step 1.2

Rewrite $\frac{x^{2}}{y^{2}}$ as ${(\frac{x}{y})}^{2}$ .

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

Step 2

Let $V = \frac{x}{y}$ . Substitute $V$ for $\frac{x}{y}$ .

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

Step 3

Solve $V = \frac{x}{y}$ for $x$ .

$x = V y$

Step 4

Use the product rule to find the derivative of $x = V y$ with respect to $y$ .

$\frac{d x}{d y} = y \frac{d V}{d y} + V$

Step 5

Substitute $y \frac{d V}{d y} + V$ for $\frac{d x}{d y}$ .

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

Step 6

Solve the substituted differential equation.

Tap for more steps...

Step 6.1

Separate the variables.

Tap for more steps...

Step 6.1.1

Solve for $\frac{d V}{d y}$ .

Tap for more steps...

Step 6.1.1.1

Move all terms not containing $\frac{d V}{d y}$ to the right side of the equation.

Tap for more steps...

Step 6.1.1.1.1

Subtract $V$ from both sides of the equation.

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

Step 6.1.1.1.2

Combine the opposite terms in $V^{2} + V + 1 - V$ .

Tap for more steps...

Step 6.1.1.1.2.1

Subtract $V$ from $V$ .

$y \frac{d V}{d y} = V^{2} + 0 + 1$

Step 6.1.1.1.2.2

Add $V^{2}$ and $0$ .

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

Step 6.1.1.2

Divide each term in $y \frac{d V}{d y} = V^{2} + 1$ by $y$ and simplify.

Tap for more steps...

Step 6.1.1.2.1

Divide each term in $y \frac{d V}{d y} = V^{2} + 1$ by $y$ .

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

Step 6.1.1.2.2

Simplify the left side.

Tap for more steps...

Step 6.1.1.2.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 6.1.1.2.2.1.1

Cancel the common factor.

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

Step 6.1.1.2.2.1.2

Divide $\frac{d V}{d y}$ by $1$ .

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

Step 6.1.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Multiply both sides by $\frac{1}{V^{2} + 1}$ .

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

Step 6.1.4

Cancel the common factor of $V^{2} + 1$ .

Tap for more steps...

Step 6.1.4.1

Cancel the common factor.

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

Step 6.1.4.2

Rewrite the expression.

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

Step 6.1.5

Rewrite the equation.

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

Step 6.2

Integrate both sides.

Tap for more steps...

Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

Tap for more steps...

Step 6.2.2.1

Simplify the expression.

Tap for more steps...

Step 6.2.2.1.1

Reorder $V^{2}$ and $1$ .

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

Step 6.2.2.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 6.2.2.2

The integral of $\frac{1}{1^{2} + V^{2}}$ with respect to $V$ is $\arctan (V) + C_{1}$ .

$\arctan (V) + C_{1} = \int \frac{1}{y} d y$

Step 6.2.3

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$\arctan (V) + C_{1} = \ln (| y |) + C_{2}$

Step 6.2.4

Group the constant of integration on the right side as $C$ .

$\arctan (V) = \ln (| y |) + C$

Step 6.3

Take the inverse arctangent of both sides of the equation to extract $V$ from inside the arctangent.

$V = \tan (\ln (| y |) + C)$

Step 7

Substitute $\frac{x}{y}$ for $V$ .

$\frac{x}{y} = \tan (\ln (| y |) + C)$

Step 8

Solve $\frac{x}{y} = \tan (\ln (| y |) + C)$ for $x$ .

Tap for more steps...

Step 8.1

Multiply both sides by $y$ .

$\frac{x}{y} y = \tan (\ln (| y |) + C) y$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Simplify the left side.

Tap for more steps...

Step 8.2.1.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 8.2.1.1.1

Cancel the common factor.

$\frac{x}{y} y = \tan (\ln (| y |) + C) y$

Step 8.2.1.1.2

Rewrite the expression.

$x = \tan (\ln (| y |) + C) y$

Step 8.2.2

Simplify the right side.

Tap for more steps...

Step 8.2.2.1

Reorder factors in $\tan (\ln (| y |) + C) y$ .

$x = y \tan (\ln (| y |) + C)$