Calculus Examples

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

Step 1

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{3}$ .

$v = y^{- 2}$

Step 2

Solve the equation for $y$ .

$y = v^{- \frac{1}{2}}$

Step 3

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- \frac{1}{2}}$

Step 4

Take the derivative of $v^{- \frac{1}{2}}$ with respect to $x$ .

Tap for more steps...

Step 4.1

Take the derivative of $v^{- \frac{1}{2}}$ .

$y' = \frac{d}{d x} [v^{- \frac{1}{2}}]$

Step 4.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{2}}}]$

Step 4.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = v^{\frac{1}{2}}$ .

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v^{\frac{1}{2}}]}{{(v^{\frac{1}{2}})}^{2}}$

Step 4.4

Simplify the expression.

Tap for more steps...

Step 4.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{{(v^{\frac{1}{2}})}^{2}}$

Step 4.4.2

Multiply the exponents in ${(v^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.4.2.1

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

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v^{\frac{1}{2} \cdot 2}}$

Step 4.4.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.2.1

Cancel the common factor.

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v^{\frac{1}{2} \cdot 2}}$

Step 4.4.2.2.2

Rewrite the expression.

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v^{1}}$

Step 4.5

Simplify.

$y' = \frac{v^{\frac{1}{2}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.6.1

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

$y' = \frac{v^{\frac{1}{2}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6.2

Simplify the expression.

Tap for more steps...

Step 4.6.2.1

Multiply $v^{\frac{1}{2}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6.2.2

Subtract $\frac{d}{d x} [v^{\frac{1}{2}}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.6.2.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{\frac{1}{2}}]}{v}$

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = v$ .

Tap for more steps...

Step 4.7.1

To apply the Chain Rule, set $u_{1}$ as $v$ .

$y' = - \frac{\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [v]}{v}$

Step 4.7.2

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

$y' = - \frac{\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [v]}{v}$

Step 4.7.3

Replace all occurrences of $u_{1}$ with $v$ .

$y' = - \frac{\frac{1}{2} v^{\frac{1}{2} - 1} \frac{d}{d x} [v]}{v}$

Step 4.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{\frac{1}{2} v^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.9

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

$y' = - \frac{\frac{1}{2} v^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.10

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{2} v^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.11

Simplify the numerator.

Tap for more steps...

Step 4.11.1

Multiply $- 1$ by $2$ .

$y' = - \frac{\frac{1}{2} v^{\frac{1 - 2}{2}} \frac{d}{d x} [v]}{v}$

Step 4.11.2

Subtract $2$ from $1$ .

$y' = - \frac{\frac{1}{2} v^{\frac{- 1}{2}} \frac{d}{d x} [v]}{v}$

Step 4.12

Move the negative in front of the fraction.

$y' = - \frac{\frac{1}{2} v^{- \frac{1}{2}} \frac{d}{d x} [v]}{v}$

Step 4.13

Combine $\frac{1}{2}$ and $v^{- \frac{1}{2}}$ .

$y' = - \frac{\frac{v^{- \frac{1}{2}}}{2} \frac{d}{d x} [v]}{v}$

Step 4.14

Move $v^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{\frac{1}{2 v^{\frac{1}{2}}} \frac{d}{d x} [v]}{v}$

Step 4.15

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$y' = - \frac{\frac{1}{2 v^{\frac{1}{2}}} v'}{v}$

Step 4.16

Combine $\frac{1}{2 v^{\frac{1}{2}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{2 v^{\frac{1}{2}}}}{v}$

Step 4.17

Rewrite $\frac{\frac{v'}{2 v^{\frac{1}{2}}}}{v}$ as a product.

$y' = - (\frac{v'}{2 v^{\frac{1}{2}}} \cdot \frac{1}{v})$

Step 4.18

Multiply $\frac{v'}{2 v^{\frac{1}{2}}}$ by $\frac{1}{v}$ .

$y' = - \frac{v'}{2 v^{\frac{1}{2}} v}$

Step 4.19

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

$y' = - \frac{v'}{2 (v^{1} v^{\frac{1}{2}})}$

Step 4.20

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

$y' = - \frac{v'}{2 v^{1 + \frac{1}{2}}}$

Step 4.21

Write $1$ as a fraction with a common denominator.

$y' = - \frac{v'}{2 v^{\frac{2}{2} + \frac{1}{2}}}$

Step 4.22

Combine the numerators over the common denominator.

$y' = - \frac{v'}{2 v^{\frac{2 + 1}{2}}}$

Step 4.23

Add $2$ and $1$ .

$y' = - \frac{v'}{2 v^{\frac{3}{2}}}$

Step 5

Substitute $- \frac{v'}{2 v^{\frac{3}{2}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{2}}$ for $y$ in the original equation $\frac{d y}{d x} - y = y^{3}$ .

$- \frac{v'}{2 v^{\frac{3}{2}}} - v^{- \frac{1}{2}} = {(v^{- \frac{1}{2}})}^{3}$

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 $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - {(v^{- \frac{1}{2}})}^{3} = 0$

Step 6.1.1.1.2

Multiply the exponents in ${(v^{- \frac{1}{2}})}^{3}$ .

Tap for more steps...

Step 6.1.1.1.2.1

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

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - v^{- \frac{1}{2} \cdot 3} = 0$

Step 6.1.1.1.2.2

Multiply $- \frac{1}{2} \cdot 3$ .

Tap for more steps...

Step 6.1.1.1.2.2.1

Multiply $3$ by $- 1$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - v^{- 3 (\frac{1}{2})} = 0$

Step 6.1.1.1.2.2.2

Combine $- 3$ and $\frac{1}{2}$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - v^{\frac{- 3}{2}} = 0$

Step 6.1.1.1.2.3

Move the negative in front of the fraction.

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - v^{- \frac{3}{2}} = 0$

Step 6.1.1.1.3

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

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} - \frac{1}{v^{\frac{1}{2}}} - \frac{1}{v^{\frac{3}{2}}} = 0$

Step 6.1.1.2

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

Tap for more steps...

Step 6.1.1.2.1

Add $\frac{1}{v^{\frac{1}{2}}}$ to both sides of the equation.

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

Step 6.1.1.2.2

Add $\frac{1}{v^{\frac{3}{2}}}$ to both sides of the equation.

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

Step 6.1.1.3

Divide each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} = \frac{1}{v^{\frac{1}{2}}} + \frac{1}{v^{\frac{3}{2}}}$ by $- 1$ and simplify.

Tap for more steps...

Step 6.1.1.3.1

Divide each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} = \frac{1}{v^{\frac{1}{2}}} + \frac{1}{v^{\frac{3}{2}}}$ by $- 1$ .

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

Step 6.1.1.3.2

Simplify the left side.

Tap for more steps...

Step 6.1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 6.1.1.3.2.2

Divide $\frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}}$ by $1$ .

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

Step 6.1.1.3.3

Simplify the right side.

Tap for more steps...

Step 6.1.1.3.3.1

Simplify each term.

Tap for more steps...

Step 6.1.1.3.3.1.1

Move the negative one from the denominator of $\frac{\frac{1}{v^{\frac{1}{2}}}}{- 1}$ .

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

Step 6.1.1.3.3.1.2

Rewrite $- 1 \cdot \frac{1}{v^{\frac{1}{2}}}$ as $- \frac{1}{v^{\frac{1}{2}}}$ .

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

Step 6.1.1.3.3.1.3

Move the negative one from the denominator of $\frac{\frac{1}{v^{\frac{3}{2}}}}{- 1}$ .

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

Step 6.1.1.3.3.1.4

Rewrite $- 1 \cdot \frac{1}{v^{\frac{3}{2}}}$ as $- \frac{1}{v^{\frac{3}{2}}}$ .

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

Step 6.1.1.4

Multiply both sides by $2 v^{\frac{3}{2}}$ .

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

Step 6.1.1.5

Simplify.

Tap for more steps...

Step 6.1.1.5.1

Simplify the left side.

Tap for more steps...

Step 6.1.1.5.1.1

Simplify $\frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}})$ .

Tap for more steps...

Step 6.1.1.5.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.1.5.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.1.1.5.1.1.2.1

Cancel the common factor.

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

Step 6.1.1.5.1.1.2.2

Rewrite the expression.

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

Step 6.1.1.5.1.1.3

Cancel the common factor of $v^{\frac{3}{2}}$ .

Tap for more steps...

Step 6.1.1.5.1.1.3.1

Cancel the common factor.

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

Step 6.1.1.5.1.1.3.2

Rewrite the expression.

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

Step 6.1.1.5.2

Simplify the right side.

Tap for more steps...

Step 6.1.1.5.2.1

Simplify $(- \frac{1}{v^{\frac{1}{2}}} - \frac{1}{v^{\frac{3}{2}}}) (2 v^{\frac{3}{2}})$ .

Tap for more steps...

Step 6.1.1.5.2.1.1

Simplify terms.

Tap for more steps...

Step 6.1.1.5.2.1.1.1

Apply the distributive property.

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

Step 6.1.1.5.2.1.1.2

Cancel the common factor of $v^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.1.1.5.2.1.1.2.1

Move the leading negative in $- \frac{1}{v^{\frac{1}{2}}}$ into the numerator.

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

Step 6.1.1.5.2.1.1.2.2

Factor $v^{\frac{1}{2}}$ out of $2 v^{\frac{3}{2}}$ .

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

Step 6.1.1.5.2.1.1.2.3

Cancel the common factor.

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

Step 6.1.1.5.2.1.1.2.4

Rewrite the expression.

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

Step 6.1.1.5.2.1.1.3

Multiply $2$ by $- 1$ .

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

Step 6.1.1.5.2.1.1.4

Cancel the common factor of $v^{\frac{3}{2}}$ .

Tap for more steps...

Step 6.1.1.5.2.1.1.4.1

Move the leading negative in $- \frac{1}{v^{\frac{3}{2}}}$ into the numerator.

$\frac{d v}{d x} = - 2 v^{\frac{2}{2}} + \frac{- 1}{v^{\frac{3}{2}}} (2 v^{\frac{3}{2}})$

Step 6.1.1.5.2.1.1.4.2

Factor $v^{\frac{3}{2}}$ out of $2 v^{\frac{3}{2}}$ .

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

Step 6.1.1.5.2.1.1.4.3

Cancel the common factor.

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

Step 6.1.1.5.2.1.1.4.4

Rewrite the expression.

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

Step 6.1.1.5.2.1.1.5

Multiply $- 1$ by $2$ .

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

Step 6.1.1.5.2.1.2

Simplify each term.

Tap for more steps...

Step 6.1.1.5.2.1.2.1

Divide $2$ by $2$ .

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

Step 6.1.1.5.2.1.2.2

Simplify.

$\frac{d v}{d x} = - 2 v - 2$

Step 6.1.2

Multiply both sides by $\frac{1}{- 2 v - 2}$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{1}{- 2 v - 2} (- 2 v - 2)$

Step 6.1.3

Simplify.

Tap for more steps...

Step 6.1.3.1

Factor $2$ out of $- 2 v - 2$ .

Tap for more steps...

Step 6.1.3.1.1

Factor $2$ out of $- 2 v$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{1}{2 (- v) - 2} (- 2 v - 2)$

Step 6.1.3.1.2

Factor $2$ out of $- 2$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{1}{2 (- v) + 2 (- 1)} (- 2 v - 2)$

Step 6.1.3.1.3

Factor $2$ out of $2 (- v) + 2 (- 1)$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{1}{2 (- v - 1)} (- 2 v - 2)$

Step 6.1.3.2

Multiply $\frac{1}{2 (- v - 1)}$ by $- 2 v - 2$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{- 2 v - 2}{2 (- v - 1)}$

Step 6.1.3.3

Cancel the common factor of $- 2 v - 2$ and $2$ .

Tap for more steps...

Step 6.1.3.3.1

Factor $2$ out of $- 2 v$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{2 (- v) - 2}{2 (- v - 1)}$

Step 6.1.3.3.2

Factor $2$ out of $- 2$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{2 (- v) + 2 \cdot - 1}{2 (- v - 1)}$

Step 6.1.3.3.3

Factor $2$ out of $2 (- v) + 2 (- 1)$ .

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{2 (- v - 1)}{2 (- v - 1)}$

Step 6.1.3.3.4

Cancel the common factors.

Tap for more steps...

Step 6.1.3.3.4.1

Cancel the common factor.

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{2 (- v - 1)}{2 (- v - 1)}$

Step 6.1.3.3.4.2

Rewrite the expression.

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{- v - 1}{- v - 1}$

Step 6.1.3.4

Cancel the common factor of $- v - 1$ .

Tap for more steps...

Step 6.1.3.4.1

Cancel the common factor.

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = \frac{- v - 1}{- v - 1}$

Step 6.1.3.4.2

Rewrite the expression.

$\frac{1}{- 2 v - 2} \frac{d v}{d x} = 1$

Step 6.1.4

Rewrite the equation.

$\frac{1}{- 2 v - 2} d v = 1 d x$

Step 6.2

Integrate both sides.

Tap for more steps...

Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{- 2 v - 2} d v = \int d x$

Step 6.2.2

Integrate the left side.

Tap for more steps...

Step 6.2.2.1

Let $u_{2} = - 2 v - 2$ . Then $d u_{2} = - 2 d v$ , so $- \frac{1}{2} d u_{2} = d v$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 6.2.2.1.1

Let $u_{2} = - 2 v - 2$ . Find $\frac{d u_{2}}{d v}$ .

Tap for more steps...

Step 6.2.2.1.1.1

Differentiate $- 2 v - 2$ .

$\frac{d}{d v} [- 2 v - 2]$

Step 6.2.2.1.1.2

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

$\frac{d}{d v} [- 2 v] + \frac{d}{d v} [- 2]$

Step 6.2.2.1.1.3

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

Tap for more steps...

Step 6.2.2.1.1.3.1

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

$- 2 \frac{d}{d v} [v] + \frac{d}{d v} [- 2]$

Step 6.2.2.1.1.3.2

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

$- 2 \cdot 1 + \frac{d}{d v} [- 2]$

Step 6.2.2.1.1.3.3

Multiply $- 2$ by $1$ .

$- 2 + \frac{d}{d v} [- 2]$

Step 6.2.2.1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 6.2.2.1.1.4.1

Since $- 2$ is constant with respect to $v$ , the derivative of $- 2$ with respect to $v$ is $0$ .

$- 2 + 0$

Step 6.2.2.1.1.4.2

Add $- 2$ and $0$ .

$- 2$

Step 6.2.2.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{u_{2}} \cdot \frac{1}{- 2} d u_{2} = \int d x$

Step 6.2.2.2

Simplify.

Tap for more steps...

Step 6.2.2.2.1

Move the negative in front of the fraction.

$\int \frac{1}{u_{2}} (- \frac{1}{2}) d u_{2} = \int d x$

Step 6.2.2.2.2

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$\int - \frac{1}{u_{2} \cdot 2} d u_{2} = \int d x$

Step 6.2.2.2.3

Move $2$ to the left of $u_{2}$ .

$\int - \frac{1}{2 u_{2}} d u_{2} = \int d x$

Step 6.2.2.3

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- \int \frac{1}{2 u_{2}} d u_{2} = \int d x$

Step 6.2.2.4

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2}) = \int d x$

Step 6.2.2.5

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- \frac{1}{2} (\ln (| u_{2} |) + C_{1}) = \int d x$

Step 6.2.2.6

Simplify.

$- \frac{1}{2} \ln (| u_{2} |) + C_{1} = \int d x$

Step 6.2.2.7

Replace all occurrences of $u_{2}$ with $- 2 v - 2$ .

$- \frac{1}{2} \ln (| - 2 v - 2 |) + C_{1} = \int d x$

Step 6.2.3

Apply the constant rule.

$- \frac{1}{2} \ln (| - 2 v - 2 |) + C_{1} = x + C_{2}$

Step 6.2.4

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

$- \frac{1}{2} \ln (| - 2 v - 2 |) = x + C$

Step 6.3

Solve for $v$ .

Tap for more steps...

Step 6.3.1

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} \ln (| - 2 v - 2 |)) = - 2 (x + C)$

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 $- 2 (- \frac{1}{2} \ln (| - 2 v - 2 |))$ .

Tap for more steps...

Step 6.3.2.1.1.1

Combine $\ln (| - 2 v - 2 |)$ and $\frac{1}{2}$ .

$- 2 (- \frac{\ln (| - 2 v - 2 |)}{2}) = - 2 (x + C)$

Step 6.3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1.1.2.1

Move the leading negative in $- \frac{\ln (| - 2 v - 2 |)}{2}$ into the numerator.

$- 2 \frac{- \ln (| - 2 v - 2 |)}{2} = - 2 (x + C)$

Step 6.3.2.1.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \ln (| - 2 v - 2 |)}{2} = - 2 (x + C)$

Step 6.3.2.1.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \ln (| - 2 v - 2 |)}{2} = - 2 (x + C)$

Step 6.3.2.1.1.2.4

Rewrite the expression.

$- - \ln (| - 2 v - 2 |) = - 2 (x + C)$

Step 6.3.2.1.1.3

Multiply.

Tap for more steps...

Step 6.3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| - 2 v - 2 |) = - 2 (x + C)$

Step 6.3.2.1.1.3.2

Multiply $\ln (| - 2 v - 2 |)$ by $1$ .

$\ln (| - 2 v - 2 |) = - 2 (x + C)$

Step 6.3.2.2

Simplify the right side.

Tap for more steps...

Step 6.3.2.2.1

Apply the distributive property.

$\ln (| - 2 v - 2 |) = - 2 x - 2 C$

Step 6.3.3

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

$e^{\ln (| - 2 v - 2 |)} = e^{- 2 x - 2 C}$

Step 6.3.4

Rewrite $\ln (| - 2 v - 2 |) = - 2 x - 2 C$ 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 $b^{y} = x$ .

$e^{- 2 x - 2 C} = | - 2 v - 2 |$

Step 6.3.5

Solve for $v$ .

Tap for more steps...

Step 6.3.5.1

Rewrite the equation as $| - 2 v - 2 | = e^{- 2 x - 2 C}$ .

$| - 2 v - 2 | = e^{- 2 x - 2 C}$

Step 6.3.5.2

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

$- 2 v - 2 = \pm e^{- 2 x - 2 C}$

Step 6.3.5.3

Add $2$ to both sides of the equation.

$- 2 v = \pm e^{- 2 x - 2 C} + 2$

Step 6.3.5.4

Divide each term in $- 2 v = \pm e^{- 2 x - 2 C} + 2$ by $- 2$ and simplify.

Tap for more steps...

Step 6.3.5.4.1

Divide each term in $- 2 v = \pm e^{- 2 x - 2 C} + 2$ by $- 2$ .

$\frac{- 2 v}{- 2} = \frac{\pm e^{- 2 x - 2 C}}{- 2} + \frac{2}{- 2}$

Step 6.3.5.4.2

Simplify the left side.

Tap for more steps...

Step 6.3.5.4.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 6.3.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 v}{- 2} = \frac{\pm e^{- 2 x - 2 C}}{- 2} + \frac{2}{- 2}$

Step 6.3.5.4.2.1.2

Divide $v$ by $1$ .

$v = \frac{\pm e^{- 2 x - 2 C}}{- 2} + \frac{2}{- 2}$

Step 6.3.5.4.3

Simplify the right side.

Tap for more steps...

Step 6.3.5.4.3.1

Simplify each term.

Tap for more steps...

Step 6.3.5.4.3.1.1

Simplify $\frac{\pm e^{- 2 x - 2 C}}{- 2}$ .

$v = \frac{\pm e^{- 2 x - 2 C}}{2} + \frac{2}{- 2}$

Step 6.3.5.4.3.1.2

Divide $2$ by $- 2$ .

$v = \frac{\pm e^{- 2 x - 2 C}}{2} - 1$

Step 6.3.5.4.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$v = \frac{\pm e^{- 2 x - 2 C}}{2} - 1 \cdot \frac{2}{2}$

Step 6.3.5.4.3.3

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

$v = \frac{\pm e^{- 2 x - 2 C}}{2} + \frac{- 1 \cdot 2}{2}$

Step 6.3.5.4.3.4

Combine the numerators over the common denominator.

$v = \frac{\pm e^{- 2 x - 2 C} - 1 \cdot 2}{2}$

Step 6.3.5.4.3.5

Multiply $- 1$ by $2$ .

$v = \frac{\pm e^{- 2 x - 2 C} - 2}{2}$

Step 6.4

Group the constant terms together.

Tap for more steps...

Step 6.4.1

Simplify the constant of integration.

$v = \frac{\pm e^{- 2 x + C} - 2}{2}$

Step 6.4.2

Rewrite $e^{- 2 x + C}$ as $e^{- 2 x} e^{C}$ .

$v = \frac{\pm e^{- 2 x} e^{C} - 2}{2}$

Step 6.4.3

Reorder $e^{- 2 x}$ and $e^{C}$ .

$v = \frac{\pm e^{C} e^{- 2 x} - 2}{2}$

Step 6.4.4

Combine constants with the plus or minus.

$v = \frac{C e^{- 2 x} - 2}{2}$

Step 7

Substitute $y^{- 2}$ for $v$ .

$y^{- 2} = \frac{C e^{- 2 x} - 2}{2}$