Calculus Examples

$\frac{d y}{d x} = \frac{x^{2} + y^{2}}{3 x y}$

Step 1

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

Tap for more steps...

Step 1.1

Split $\frac{x^{2} + y^{2}}{3 x y}$ and simplify.

Tap for more steps...

Step 1.1.1

Split the fraction $\frac{x^{2} + y^{2}}{3 x y}$ into two fractions.

$\frac{d y}{d x} = \frac{x^{2}}{3 x y} + \frac{y^{2}}{3 x y}$

Step 1.1.2

Simplify each term.

Tap for more steps...

Step 1.1.2.1

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

Tap for more steps...

Step 1.1.2.1.1

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

$\frac{d y}{d x} = \frac{x \cdot x}{3 x y} + \frac{y^{2}}{3 x y}$

Step 1.1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.1.2.1

Factor $x$ out of $3 x y$ .

$\frac{d y}{d x} = \frac{x \cdot x}{x (3 y)} + \frac{y^{2}}{3 x y}$

Step 1.1.2.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x \cdot x}{x (3 y)} + \frac{y^{2}}{3 x y}$

Step 1.1.2.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{x}{3 y} + \frac{y^{2}}{3 x y}$

Step 1.1.2.2

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

Tap for more steps...

Step 1.1.2.2.1

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

$\frac{d y}{d x} = \frac{x}{3 y} + \frac{y \cdot y}{3 x y}$

Step 1.1.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.2.2.1

Factor $y$ out of $3 x y$ .

$\frac{d y}{d x} = \frac{x}{3 y} + \frac{y \cdot y}{y (3 x)}$

Step 1.1.2.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x}{3 y} + \frac{y \cdot y}{y (3 x)}$

Step 1.1.2.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{x}{3 y} + \frac{y}{3 x}$

Step 1.2

Rewrite the differential equation as $f (\frac{y}{x})$ .

Tap for more steps...

Step 1.2.1

Factor out $\frac{x}{y}$ from $\frac{x}{3 y}$ .

Tap for more steps...

Step 1.2.1.1

Factor $\frac{x}{y}$ out of $\frac{x}{3 y}$ .

$\frac{d y}{d x} = \frac{x}{y} \cdot \frac{1}{3} + \frac{y}{3 x}$

Step 1.2.1.2

Reorder $\frac{x}{y}$ and $\frac{1}{3}$ .

$\frac{d y}{d x} = \frac{1}{3} \cdot \frac{x}{y} + \frac{y}{3 x}$

Step 1.2.2

Rewrite $\frac{x}{y}$ as ${(\frac{y}{x})}^{- 1}$ .

$\frac{d y}{d x} = \frac{1}{3} {(\frac{y}{x})}^{- 1} + \frac{y}{3 x}$

Step 1.3

Factor out $\frac{y}{x}$ from $\frac{y}{3 x}$ .

Tap for more steps...

Step 1.3.1

Factor $\frac{y}{x}$ out of $\frac{y}{3 x}$ .

$\frac{d y}{d x} = \frac{1}{3} {(\frac{y}{x})}^{- 1} + \frac{y}{x} \cdot \frac{1}{3}$

Step 1.3.2

Reorder $\frac{y}{x}$ and $\frac{1}{3}$ .

$\frac{d y}{d x} = \frac{1}{3} {(\frac{y}{x})}^{- 1} + \frac{1}{3} \cdot \frac{y}{x}$

Step 2

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

$\frac{d y}{d x} = \frac{1}{3} V^{- 1} + \frac{1}{3} V$

Step 3

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

$y = V x$

Step 4

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

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

Step 5

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

$x \frac{d V}{d x} + V = \frac{1}{3} V^{- 1} + \frac{1}{3} V$

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 $\frac{d V}{d x}$ .

Tap for more steps...

Step 6.1.1.1

Simplify each term.

Tap for more steps...

Step 6.1.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x \frac{d V}{d x} + V = \frac{1}{3} \cdot \frac{1}{V} + \frac{1}{3} V$

Step 6.1.1.1.2

Multiply $\frac{1}{3}$ by $\frac{1}{V}$ .

$x \frac{d V}{d x} + V = \frac{1}{3 V} + \frac{1}{3} V$

Step 6.1.1.1.3

Combine $\frac{1}{3}$ and $V$ .

$x \frac{d V}{d x} + V = \frac{1}{3 V} + \frac{V}{3}$

Step 6.1.1.2

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = \frac{1}{3 V} + \frac{V}{3} - V$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{1}{3 V} + \frac{V}{3} - V$ by $x$ and simplify.

Tap for more steps...

Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{1}{3 V} + \frac{V}{3} - V$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{3 V}}{x} + \frac{\frac{V}{3}}{x} + \frac{- V}{x}$

Step 6.1.1.3.2

Simplify the left side.

Tap for more steps...

Step 6.1.1.3.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{3 V}}{x} + \frac{\frac{V}{3}}{x} + \frac{- V}{x}$

Step 6.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{\frac{1}{3 V}}{x} + \frac{\frac{V}{3}}{x} + \frac{- V}{x}$

Step 6.1.1.3.3

Simplify the right side.

Tap for more steps...

Step 6.1.1.3.3.1

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V}{3} - V}{x}$

Step 6.1.1.3.3.2

To write $- V$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V}{3} - V \cdot \frac{3}{3}}{x}$

Step 6.1.1.3.3.3

Simplify terms.

Tap for more steps...

Step 6.1.1.3.3.3.1

Combine $- V$ and $\frac{3}{3}$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V}{3} + \frac{- V \cdot 3}{3}}{x}$

Step 6.1.1.3.3.3.2

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V - V \cdot 3}{3}}{x}$

Step 6.1.1.3.3.3.3

Simplify each term.

Tap for more steps...

Step 6.1.1.3.3.3.3.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1.3.3.3.3.1.1

Factor $V$ out of $V - V \cdot 3$ .

Tap for more steps...

Step 6.1.1.3.3.3.3.1.1.1

Raise $V$ to the power of $1$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V^{1} - V \cdot 3}{3}}{x}$

Step 6.1.1.3.3.3.3.1.1.2

Factor $V$ out of $V^{1}$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V \cdot 1 - V \cdot 3}{3}}{x}$

Step 6.1.1.3.3.3.3.1.1.3

Factor $V$ out of $- V \cdot 3$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V \cdot 1 + V (- 1 \cdot 3)}{3}}{x}$

Step 6.1.1.3.3.3.3.1.1.4

Factor $V$ out of $V \cdot 1 + V (- 1 \cdot 3)$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V (1 - 1 \cdot 3)}{3}}{x}$

Step 6.1.1.3.3.3.3.1.2

Multiply $- 1$ by $3$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V (1 - 3)}{3}}{x}$

Step 6.1.1.3.3.3.3.1.3

Subtract $3$ from $1$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{V \cdot - 2}{3}}{x}$

Step 6.1.1.3.3.3.3.2

Move $- 2$ to the left of $V$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} + \frac{- 2 \cdot V}{3}}{x}$

Step 6.1.1.3.3.3.3.3

Move the negative in front of the fraction.

$\frac{d V}{d x} = \frac{\frac{1}{3 V} - \frac{2 V}{3}}{x}$

Step 6.1.1.3.3.4

Simplify the numerator.

Tap for more steps...

Step 6.1.1.3.3.4.1

To write $- \frac{2 V}{3}$ as a fraction with a common denominator, multiply by $\frac{V}{V}$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} - \frac{2 V}{3} \cdot \frac{V}{V}}{x}$

Step 6.1.1.3.3.4.2

Multiply $\frac{2 V}{3}$ by $\frac{V}{V}$ .

$\frac{d V}{d x} = \frac{\frac{1}{3 V} - \frac{2 V \cdot V}{3 V}}{x}$

Step 6.1.1.3.3.4.3

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{\frac{1 - 2 V \cdot V}{3 V}}{x}$

Step 6.1.1.3.3.4.4

Multiply $V$ by $V$ by adding the exponents.

Tap for more steps...

Step 6.1.1.3.3.4.4.1

Move $V$ .

$\frac{d V}{d x} = \frac{\frac{1 - 2 (V \cdot V)}{3 V}}{x}$

Step 6.1.1.3.3.4.4.2

Multiply $V$ by $V$ .

$\frac{d V}{d x} = \frac{\frac{1 - 2 V^{2}}{3 V}}{x}$

Step 6.1.1.3.3.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = \frac{1 - 2 V^{2}}{3 V} \cdot \frac{1}{x}$

Step 6.1.1.3.3.6

Multiply $\frac{1 - 2 V^{2}}{3 V}$ by $\frac{1}{x}$ .

$\frac{d V}{d x} = \frac{1 - 2 V^{2}}{3 V x}$

Step 6.1.2

Regroup factors.

$\frac{d V}{d x} = \frac{1 - 2 V^{2}}{3 V} \cdot \frac{1}{x}$

Step 6.1.3

Multiply both sides by $\frac{3 V}{1 - 2 V^{2}}$ .

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{3 V}{1 - 2 V^{2}} (\frac{1 - 2 V^{2}}{3 V} \cdot \frac{1}{x})$

Step 6.1.4

Simplify.

Tap for more steps...

Step 6.1.4.1

Multiply $\frac{1 - 2 V^{2}}{3 V}$ by $\frac{1}{x}$ .

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{3 V}{1 - 2 V^{2}} \cdot \frac{1 - 2 V^{2}}{3 V x}$

Step 6.1.4.2

Cancel the common factor of $3 V$ .

Tap for more steps...

Step 6.1.4.2.1

Factor $3 V$ out of $3 V x$ .

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{3 V}{1 - 2 V^{2}} \cdot \frac{1 - 2 V^{2}}{3 V (x)}$

Step 6.1.4.2.2

Cancel the common factor.

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{3 V}{1 - 2 V^{2}} \cdot \frac{1 - 2 V^{2}}{3 V x}$

Step 6.1.4.2.3

Rewrite the expression.

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{1}{1 - 2 V^{2}} \cdot \frac{1 - 2 V^{2}}{x}$

Step 6.1.4.3

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

Tap for more steps...

Step 6.1.4.3.1

Cancel the common factor.

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{1}{1 - 2 V^{2}} \cdot \frac{1 - 2 V^{2}}{x}$

Step 6.1.4.3.2

Rewrite the expression.

$\frac{3 V}{1 - 2 V^{2}} \frac{d V}{d x} = \frac{1}{x}$

Step 6.1.5

Rewrite the equation.

$\frac{3 V}{1 - 2 V^{2}} d V = \frac{1}{x} d x$

Step 6.2

Integrate both sides.

Tap for more steps...

Step 6.2.1

Set up an integral on each side.

$\int \frac{3 V}{1 - 2 V^{2}} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

Tap for more steps...

Step 6.2.2.1

Since $3$ is constant with respect to $V$ , move $3$ out of the integral.

$3 \int \frac{V}{1 - 2 V^{2}} d V = \int \frac{1}{x} d x$

Step 6.2.2.2

Let $u = 1 - 2 V^{2}$ . Then $d u = - 4 V d V$ , so $- \frac{1}{4} d u = V d V$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 6.2.2.2.1

Let $u = 1 - 2 V^{2}$ . Find $\frac{d u}{d V}$ .

Tap for more steps...

Step 6.2.2.2.1.1

Differentiate $1 - 2 V^{2}$ .

$\frac{d}{d V} [1 - 2 V^{2}]$

Step 6.2.2.2.1.2

Differentiate.

Tap for more steps...

Step 6.2.2.2.1.2.1

By the Sum Rule, the derivative of $1 - 2 V^{2}$ with respect to $V$ is $\frac{d}{d V} [1] + \frac{d}{d V} [- 2 V^{2}]$ .

$\frac{d}{d V} [1] + \frac{d}{d V} [- 2 V^{2}]$

Step 6.2.2.2.1.2.2

Since $1$ is constant with respect to $V$ , the derivative of $1$ with respect to $V$ is $0$ .

$0 + \frac{d}{d V} [- 2 V^{2}]$

Step 6.2.2.2.1.3

Evaluate $\frac{d}{d V} [- 2 V^{2}]$ .

Tap for more steps...

Step 6.2.2.2.1.3.1

Since $- 2$ is constant with respect to $V$ , the derivative of $- 2 V^{2}$ with respect to $V$ is $- 2 \frac{d}{d V} [V^{2}]$ .

$0 - 2 \frac{d}{d V} [V^{2}]$

Step 6.2.2.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d V} [V^{n}]$ is $n V^{n - 1}$ where $n = 2$ .

$0 - 2 (2 V)$

Step 6.2.2.2.1.3.3

Multiply $2$ by $- 2$ .

$0 - 4 V$

Step 6.2.2.2.1.4

Subtract $4 V$ from $0$ .

$- 4 V$

Step 6.2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$3 \int \frac{1}{u} \cdot \frac{1}{- 4} d u = \int \frac{1}{x} d x$

Step 6.2.2.3

Simplify.

Tap for more steps...

Step 6.2.2.3.1

Move the negative in front of the fraction.

$3 \int \frac{1}{u} (- \frac{1}{4}) d u = \int \frac{1}{x} d x$

Step 6.2.2.3.2

Multiply $\frac{1}{u}$ by $\frac{1}{4}$ .

$3 \int - \frac{1}{u \cdot 4} d u = \int \frac{1}{x} d x$

Step 6.2.2.3.3

Move $4$ to the left of $u$ .

$3 \int - \frac{1}{4 u} d u = \int \frac{1}{x} d x$

Step 6.2.2.4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$3 (- \int \frac{1}{4 u} d u) = \int \frac{1}{x} d x$

Step 6.2.2.5

Multiply $- 1$ by $3$ .

$- 3 \int \frac{1}{4 u} d u = \int \frac{1}{x} d x$

Step 6.2.2.6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- 3 (\frac{1}{4} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 6.2.2.7

Simplify.

Tap for more steps...

Step 6.2.2.7.1

Combine $\frac{1}{4}$ and $- 3$ .

$\frac{- 3}{4} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.7.2

Move the negative in front of the fraction.

$- \frac{3}{4} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.8

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

$- \frac{3}{4} (\ln (| u |) + C_{1}) = \int \frac{1}{x} d x$

Step 6.2.2.9

Simplify.

$- \frac{3}{4} \ln (| u |) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.2.10

Replace all occurrences of $u$ with $1 - 2 V^{2}$ .

$- \frac{3}{4} \ln (| 1 - 2 V^{2} |) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.3

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

$- \frac{3}{4} \ln (| 1 - 2 V^{2} |) + C_{1} = \ln (| x |) + C_{2}$

Step 6.2.4

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

$- \frac{3}{4} \ln (| 1 - 2 V^{2} |) = \ln (| x |) + C$

Step 6.3

Solve for $V$ .

Tap for more steps...

Step 6.3.1

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3}{4} \ln (| 1 - 2 V^{2} |)) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2

Simplify both sides of the equation.

Tap for more steps...

Step 6.3.2.1

Simplify the left side.

Tap for more steps...

Step 6.3.2.1.1

Simplify $- \frac{4}{3} (- \frac{3}{4} \ln (| 1 - 2 V^{2} |))$ .

Tap for more steps...

Step 6.3.2.1.1.1

Combine $\ln (| 1 - 2 V^{2} |)$ and $\frac{3}{4}$ .

$- \frac{4}{3} (- \frac{\ln (| 1 - 2 V^{2} |) \cdot 3}{4}) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.2.1.1.2.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{\ln (| 1 - 2 V^{2} |) \cdot 3}{4}) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.2.2

Move the leading negative in $- \frac{\ln (| 1 - 2 V^{2} |) \cdot 3}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- \ln (| 1 - 2 V^{2} |) \cdot 3}{4} = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.2.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- \ln (| 1 - 2 V^{2} |) \cdot 3}{4} = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.2.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- \ln (| 1 - 2 V^{2} |) \cdot 3}{4} = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.2.5

Rewrite the expression.

$\frac{- 1}{3} (- \ln (| 1 - 2 V^{2} |) \cdot 3) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.2.1.1.3.1

Factor $3$ out of $- \ln (| 1 - 2 V^{2} |) \cdot 3$ .

$\frac{- 1}{3} (3 \cdot (- \ln (| 1 - 2 V^{2} |))) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.3.2

Cancel the common factor.

$\frac{- 1}{3} (3 \cdot (- \ln (| 1 - 2 V^{2} |))) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.3.3

Rewrite the expression.

$- - \ln (| 1 - 2 V^{2} |) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.4

Multiply.

Tap for more steps...

Step 6.3.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| 1 - 2 V^{2} |) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.1.1.4.2

Multiply $\ln (| 1 - 2 V^{2} |)$ by $1$ .

$\ln (| 1 - 2 V^{2} |) = - \frac{4}{3} (\ln (| x |) + C)$

Step 6.3.2.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.2.1

Simplify $- \frac{4}{3} (\ln (| x |) + C)$ .

Tap for more steps...

Step 6.3.2.2.1.1

Simplify terms.

Tap for more steps...

Step 6.3.2.2.1.1.1

Apply the distributive property.

$\ln (| 1 - 2 V^{2} |) = - \frac{4}{3} \ln (| x |) - \frac{4}{3} C$

Step 6.3.2.2.1.1.2

Combine $\ln (| x |)$ and $\frac{4}{3}$ .

$\ln (| 1 - 2 V^{2} |) = - \frac{\ln (| x |) \cdot 4}{3} - \frac{4}{3} C$

Step 6.3.2.2.1.1.3

Combine $C$ and $\frac{4}{3}$ .

$\ln (| 1 - 2 V^{2} |) = - \frac{\ln (| x |) \cdot 4}{3} - \frac{C \cdot 4}{3}$

Step 6.3.2.2.1.2

Simplify each term.

Tap for more steps...

Step 6.3.2.2.1.2.1

Move $4$ to the left of $\ln (| x |)$ .

$\ln (| 1 - 2 V^{2} |) = - \frac{4 \cdot \ln (| x |)}{3} - \frac{C \cdot 4}{3}$

Step 6.3.2.2.1.2.2

Move $4$ to the left of $C$ .

$\ln (| 1 - 2 V^{2} |) = - \frac{4 \ln (| x |)}{3} - \frac{4 C}{3}$

Step 6.3.3

Multiply each term in $\ln (| 1 - 2 V^{2} |) = - \frac{4 \ln (| x |)}{3} - \frac{4 C}{3}$ by $3$ to eliminate the fractions.

Tap for more steps...

Step 6.3.3.1

Multiply each term in $\ln (| 1 - 2 V^{2} |) = - \frac{4 \ln (| x |)}{3} - \frac{4 C}{3}$ by $3$ .

$\ln (| 1 - 2 V^{2} |) \cdot 3 = - \frac{4 \ln (| x |)}{3} \cdot 3 - \frac{4 C}{3} \cdot 3$

Step 6.3.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.3.2.1

Move $3$ to the left of $\ln (| 1 - 2 V^{2} |)$ .

$3 \ln (| 1 - 2 V^{2} |) = - \frac{4 \ln (| x |)}{3} \cdot 3 - \frac{4 C}{3} \cdot 3$

Step 6.3.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.3.1

Simplify each term.

Tap for more steps...

Step 6.3.3.3.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.3.3.1.1.1

Move the leading negative in $- \frac{4 \ln (| x |)}{3}$ into the numerator.

$3 \ln (| 1 - 2 V^{2} |) = \frac{- 4 \ln (| x |)}{3} \cdot 3 - \frac{4 C}{3} \cdot 3$

Step 6.3.3.3.1.1.2

Cancel the common factor.

$3 \ln (| 1 - 2 V^{2} |) = \frac{- 4 \ln (| x |)}{3} \cdot 3 - \frac{4 C}{3} \cdot 3$

Step 6.3.3.3.1.1.3

Rewrite the expression.

$3 \ln (| 1 - 2 V^{2} |) = - 4 \ln (| x |) - \frac{4 C}{3} \cdot 3$

Step 6.3.3.3.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.3.3.1.2.1

Move the leading negative in $- \frac{4 C}{3}$ into the numerator.

$3 \ln (| 1 - 2 V^{2} |) = - 4 \ln (| x |) + \frac{- 4 C}{3} \cdot 3$

Step 6.3.3.3.1.2.2

Cancel the common factor.

$3 \ln (| 1 - 2 V^{2} |) = - 4 \ln (| x |) + \frac{- 4 C}{3} \cdot 3$

Step 6.3.3.3.1.2.3

Rewrite the expression.

$3 \ln (| 1 - 2 V^{2} |) = - 4 \ln (| x |) - 4 C$

Step 6.3.4

Move all the terms containing a logarithm to the left side of the equation.

$3 \ln (| 1 - 2 V^{2} |) + 4 \ln (| x |) = - 4 C$

Step 6.3.5

Simplify the left side.

Tap for more steps...

Step 6.3.5.1

Simplify $3 \ln (| 1 - 2 V^{2} |) + 4 \ln (| x |)$ .

Tap for more steps...

Step 6.3.5.1.1

Simplify each term.

Tap for more steps...

Step 6.3.5.1.1.1

Simplify $3 \ln (| 1 - 2 V^{2} |)$ by moving $3$ inside the logarithm.

$\ln ({| 1 - 2 V^{2} |}^{3}) + 4 \ln (| x |) = - 4 C$

Step 6.3.5.1.1.2

Simplify $4 \ln (| x |)$ by moving $4$ inside the logarithm.

$\ln ({| 1 - 2 V^{2} |}^{3}) + \ln ({| x |}^{4}) = - 4 C$

Step 6.3.5.1.1.3

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$\ln ({| 1 - 2 V^{2} |}^{3}) + \ln (x^{4}) = - 4 C$

Step 6.3.5.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ({| 1 - 2 V^{2} |}^{3} x^{4}) = - 4 C$

Step 6.3.5.1.3

Reorder factors in $\ln ({| 1 - 2 V^{2} |}^{3} x^{4})$ .

$\ln (x^{4} {| 1 - 2 V^{2} |}^{3}) = - 4 C$

Step 6.3.6

To solve for $V$ , rewrite the equation using properties of logarithms.

$e^{\ln (x^{4} {| 1 - 2 V^{2} |}^{3})} = e^{- 4 C}$

Step 6.3.7

Rewrite $\ln (x^{4} {| 1 - 2 V^{2} |}^{3}) = - 4 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 4 C} = x^{4} {| 1 - 2 V^{2} |}^{3}$

Step 6.3.8

Solve for $V$ .

Tap for more steps...

Step 6.3.8.1

Rewrite the equation as $x^{4} {| 1 - 2 V^{2} |}^{3} = e^{- 4 C}$ .

$x^{4} {| 1 - 2 V^{2} |}^{3} = e^{- 4 C}$

Step 6.3.8.2

Divide each term in $x^{4} {| 1 - 2 V^{2} |}^{3} = e^{- 4 C}$ by $x^{4}$ and simplify.

Tap for more steps...

Step 6.3.8.2.1

Divide each term in $x^{4} {| 1 - 2 V^{2} |}^{3} = e^{- 4 C}$ by $x^{4}$ .

$\frac{x^{4} {| 1 - 2 V^{2} |}^{3}}{x^{4}} = \frac{e^{- 4 C}}{x^{4}}$

Step 6.3.8.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.8.2.2.1

Cancel the common factor of $x^{4}$ .

Tap for more steps...

Step 6.3.8.2.2.1.1

Cancel the common factor.

$\frac{x^{4} {| 1 - 2 V^{2} |}^{3}}{x^{4}} = \frac{e^{- 4 C}}{x^{4}}$

Step 6.3.8.2.2.1.2

Divide ${| 1 - 2 V^{2} |}^{3}$ by $1$ .

${| 1 - 2 V^{2} |}^{3} = \frac{e^{- 4 C}}{x^{4}}$

Step 6.3.8.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$| 1 - 2 V^{2} | = \sqrt[3]{\frac{e^{- 4 C}}{x^{4}}}$

Step 6.3.8.4

Simplify $\sqrt[3]{\frac{e^{- 4 C}}{x^{4}}}$ .

Tap for more steps...

Step 6.3.8.4.1

Rewrite $\frac{e^{- 4 C}}{x^{4}}$ as ${(\frac{1}{x})}^{3} \frac{e^{- 4 C}}{x}$ .

Tap for more steps...

Step 6.3.8.4.1.1

Factor the perfect power $1^{3}$ out of $e^{- 4 C}$ .

$| 1 - 2 V^{2} | = \sqrt[3]{\frac{1^{3} e^{- 4 C}}{x^{4}}}$

Step 6.3.8.4.1.2

Factor the perfect power $x^{3}$ out of $x^{4}$ .

$| 1 - 2 V^{2} | = \sqrt[3]{\frac{1^{3} e^{- 4 C}}{x^{3} x}}$

Step 6.3.8.4.1.3

Rearrange the fraction $\frac{1^{3} e^{- 4 C}}{x^{3} x}$ .

$| 1 - 2 V^{2} | = \sqrt[3]{{(\frac{1}{x})}^{3} \frac{e^{- 4 C}}{x}}$

Step 6.3.8.4.2

Pull terms out from under the radical.

$| 1 - 2 V^{2} | = \frac{1}{x} \sqrt[3]{\frac{e^{- 4 C}}{x}}$

Step 6.3.8.4.3

Rewrite $\sqrt[3]{\frac{e^{- 4 C}}{x}}$ as $\frac{\sqrt[3]{e^{- 4 C}}}{\sqrt[3]{x}}$ .

$| 1 - 2 V^{2} | = \frac{1}{x} \cdot \frac{\sqrt[3]{e^{- 4 C}}}{\sqrt[3]{x}}$

Step 6.3.8.4.4

Combine.

$| 1 - 2 V^{2} | = \frac{1 \sqrt[3]{e^{- 4 C}}}{x \sqrt[3]{x}}$

Step 6.3.8.4.5

Multiply $\sqrt[3]{e^{- 4 C}}$ by $1$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}}}{x \sqrt[3]{x}}$

Step 6.3.8.4.6

Multiply $\frac{\sqrt[3]{e^{- 4 C}}}{x \sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}}}{x \sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Step 6.3.8.4.7

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.8.4.7.1

Multiply $\frac{\sqrt[3]{e^{- 4 C}}}{x \sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 6.3.8.4.7.2

Move $\sqrt[3]{x}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x (\sqrt[3]{x} {\sqrt[3]{x}}^{2})}$

Step 6.3.8.4.7.3

Raise $\sqrt[3]{x}$ to the power of $1$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x ({\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2})}$

Step 6.3.8.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x {\sqrt[3]{x}}^{1 + 2}}$

Step 6.3.8.4.7.5

Add $1$ and $2$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x {\sqrt[3]{x}}^{3}}$

Step 6.3.8.4.7.6

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

Tap for more steps...

Step 6.3.8.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x {(x^{\frac{1}{3}})}^{3}}$

Step 6.3.8.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \cdot x^{\frac{1}{3} \cdot 3}}$

Step 6.3.8.4.7.6.3

Combine $\frac{1}{3}$ and $3$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \cdot x^{\frac{3}{3}}}$

Step 6.3.8.4.7.6.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.8.4.7.6.4.1

Cancel the common factor.

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \cdot x^{\frac{3}{3}}}$

Step 6.3.8.4.7.6.4.2

Rewrite the expression.

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \cdot x^{1}}$

Step 6.3.8.4.7.6.5

Simplify.

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x \cdot x}$

Step 6.3.8.4.8

Multiply $x$ by $x$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} {\sqrt[3]{x}}^{2}}{x^{2}}$

Step 6.3.8.4.9

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C}} \sqrt[3]{x^{2}}}{x^{2}}$

Step 6.3.8.4.10

Combine using the product rule for radicals.

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{e^{- 4 C} x^{2}}}{x^{2}}$

Step 6.3.8.4.11

Reorder factors in $\frac{\sqrt[3]{e^{- 4 C} x^{2}}}{x^{2}}$ .

$| 1 - 2 V^{2} | = \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}$

Step 6.3.8.5

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$1 - 2 V^{2} = \pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}$

Step 6.3.8.6

Subtract $1$ from both sides of the equation.

$- 2 V^{2} = \pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} - 1$

Step 6.3.8.7

Divide each term in $- 2 V^{2} = \pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} - 1$ by $- 2$ and simplify.

Tap for more steps...

Step 6.3.8.7.1

Divide each term in $- 2 V^{2} = \pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} - 1$ by $- 2$ .

$\frac{- 2 V^{2}}{- 2} = \frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{- 2} + \frac{- 1}{- 2}$

Step 6.3.8.7.2

Simplify the left side.

Tap for more steps...

Step 6.3.8.7.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 6.3.8.7.2.1.1

Cancel the common factor.

$\frac{- 2 V^{2}}{- 2} = \frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{- 2} + \frac{- 1}{- 2}$

Step 6.3.8.7.2.1.2

Divide $V^{2}$ by $1$ .

$V^{2} = \frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{- 2} + \frac{- 1}{- 2}$

Step 6.3.8.7.3

Simplify the right side.

Tap for more steps...

Step 6.3.8.7.3.1

Simplify each term.

Tap for more steps...

Step 6.3.8.7.3.1.1

Simplify $\frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{- 2}$ .

$V^{2} = \frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{2} + \frac{- 1}{- 2}$

Step 6.3.8.7.3.1.2

Dividing two negative values results in a positive value.

$V^{2} = \frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{2} + \frac{1}{2}$

Step 6.3.8.8

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$V = \pm \sqrt{\frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{2} + \frac{1}{2}}$

Step 6.3.8.9

Simplify $\pm \sqrt{\frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}}}{2} + \frac{1}{2}}$ .

Tap for more steps...

Step 6.3.8.9.1

Combine the numerators over the common denominator.

$V = \pm \sqrt{\frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}{2}}$

Step 6.3.8.9.2

Rewrite $\sqrt{\frac{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}{2}}$ as $\frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}}{\sqrt{2}}$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}}{\sqrt{2}}$

Step 6.3.8.9.3

Multiply $\frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3.8.9.4

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.8.9.4.1

Multiply $\frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.8.9.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.3.8.9.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.3.8.9.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.3.8.9.4.5

Add $1$ and $1$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.3.8.9.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 6.3.8.9.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.3.8.9.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.3.8.9.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.8.9.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.8.9.4.6.4.1

Cancel the common factor.

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.8.9.4.6.4.2

Rewrite the expression.

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{2^{1}}$

Step 6.3.8.9.4.6.5

Evaluate the exponent.

$V = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1} \sqrt{2}}{2}$

Step 6.3.8.9.5

Combine using the product rule for radicals.

$V = \pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} e^{- 4 C}}}{x^{2}} + 1) \cdot 2}}{2}$

Step 6.4

Simplify the constant of integration.

$V = \pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}$

Step 7

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

$\frac{y}{x} = \pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}$

Step 8

Solve $\frac{y}{x} = \pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}$ for $y$ .

Tap for more steps...

Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = (\pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}) x$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{y}{x} x = (\pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}) x$

Step 8.2.1.2

Rewrite the expression.

$y = (\pm \frac{\sqrt{(\pm \frac{\sqrt[3]{x^{2} C}}{x^{2}} + 1) \cdot 2}}{2}) x$