Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Divide each term in $x \frac{d y}{d x} + y = x y^{3}$ by $x$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Divide $y^{3}$ by $1$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

$v = y^{- 2}$

Step 3

Solve the equation for $y$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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 5.3

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 5.4

Simplify the expression.

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Step 5.4.1

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 5.4.2

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

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Step 5.4.2.1

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 5.4.2.2

Cancel the common factor of $2$ .

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Step 5.4.2.2.1

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 5.4.2.2.2

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 5.5

Simplify.

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

Step 5.6

Differentiate using the Constant Rule.

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Step 5.6.1

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 5.6.2

Simplify the expression.

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Step 5.6.2.1

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

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

Step 5.6.2.2

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

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

Step 5.6.2.3

Move the negative in front of the fraction.

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

Step 5.7

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 $g (x) = v$ .

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Step 5.7.1

To apply the Chain Rule, set $u$ as $v$ .

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

Step 5.7.2

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

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

Step 5.7.3

Replace all occurrences of $u$ with $v$ .

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

Step 5.8

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 5.9

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 5.10

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 5.11

Simplify the numerator.

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Step 5.11.1

Multiply $- 1$ by $2$ .

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

Step 5.11.2

Subtract $2$ from $1$ .

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

Step 5.12

Move the negative in front of the fraction.

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

Step 5.13

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

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

Step 5.14

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 5.15

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

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

Step 5.16

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

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

Step 5.17

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 5.18

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

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

Step 5.19

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

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

Step 5.20

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

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

Step 5.21

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

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

Step 5.22

Combine the numerators over the common denominator.

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

Step 5.23

Add $2$ and $1$ .

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

Step 6

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

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

Step 7

Solve the substituted differential equation.

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Step 7.1

Rewrite the differential equation as $\frac{d v}{d x} + M (x) v = Q (x)$ .

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Step 7.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} + \frac{1}{x} v^{- \frac{1}{2}} = {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ to eliminate the fractions.

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Step 7.1.1.1

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

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

Step 7.1.1.2

Simplify the left side.

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Step 7.1.1.2.1

Simplify each term.

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Step 7.1.1.2.1.1

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

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Step 7.1.1.2.1.1.1

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

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

Step 7.1.1.2.1.1.2

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

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

Step 7.1.1.2.1.1.3

Cancel the common factor.

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

Step 7.1.1.2.1.1.4

Rewrite the expression.

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

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

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

Step 7.1.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 7.1.1.2.1.5

Multiply $v^{- \frac{1}{2}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

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Step 7.1.1.2.1.5.1

Move $v^{\frac{3}{2}}$ .

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

Step 7.1.1.2.1.5.2

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

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

Step 7.1.1.2.1.5.3

Combine the numerators over the common denominator.

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

Step 7.1.1.2.1.5.4

Subtract $1$ from $3$ .

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

Step 7.1.1.2.1.5.5

Divide $2$ by $2$ .

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

Step 7.1.1.2.1.6

Simplify $- 1 \cdot 2 \frac{1}{x} v^{1}$ .

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

Step 7.1.1.2.1.7

Multiply $- 1$ by $2$ .

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

Step 7.1.1.2.1.8

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

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

Step 7.1.1.2.1.9

Move the negative in front of the fraction.

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

Step 7.1.1.2.1.10

Combine $v$ and $\frac{2}{x}$ .

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

Step 7.1.1.2.1.11

Move $2$ to the left of $v$ .

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

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 7.1.1.3.2

Multiply $- 1$ by $2$ .

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

Step 7.1.1.3.3

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

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Step 7.1.1.3.3.1

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

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

Step 7.1.1.3.3.2

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

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Step 7.1.1.3.3.2.1

Multiply $3$ by $- 1$ .

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

Step 7.1.1.3.3.2.2

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

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

Step 7.1.1.3.3.3

Move the negative in front of the fraction.

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

Step 7.1.1.3.4

Multiply $v^{- \frac{3}{2}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

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Step 7.1.1.3.4.1

Move $v^{\frac{3}{2}}$ .

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

Step 7.1.1.3.4.2

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

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

Step 7.1.1.3.4.3

Combine the numerators over the common denominator.

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

Step 7.1.1.3.4.4

Subtract $3$ from $3$ .

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

Step 7.1.1.3.4.5

Divide $0$ by $2$ .

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

Step 7.1.1.3.5

Simplify $- 2 v^{0}$ .

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

Step 7.1.2

Factor $v$ out of $- \frac{2 v}{x}$ .

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

Step 7.1.3

Reorder $v$ and $- \frac{2}{x}$ .

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

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{2}{x}$ .

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Step 7.2.1

Set up the integration.

$e^{\int - \frac{2}{x} d x}$

Step 7.2.2

Integrate $- \frac{2}{x}$ .

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Step 7.2.2.1

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

$e^{- \int \frac{2}{x} d x}$

Step 7.2.2.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{- (2 \int \frac{1}{x} d x)}$

Step 7.2.2.3

Multiply $2$ by $- 1$ .

$e^{- 2 \int \frac{1}{x} d x}$

Step 7.2.2.4

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

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

Step 7.2.2.5

Simplify.

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

Step 7.2.3

Remove the constant of integration.

$e^{- 2 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 2})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{- 2}$

Step 7.2.6

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

$\frac{1}{x^{2}}$

Step 7.3

Multiply each term by the integrating factor $\frac{1}{x^{2}}$ .

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Step 7.3.1

Multiply each term by $\frac{1}{x^{2}}$ .

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.3.2.3

Combine $\frac{2}{x}$ and $v$ .

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

Step 7.3.2.4

Multiply $- \frac{1}{x^{2}} \cdot \frac{2 v}{x}$ .

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Step 7.3.2.4.1

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

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

Step 7.3.2.4.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.3.2.4.2.1

Multiply $x$ by $x^{2}$ .

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Step 7.3.2.4.2.1.1

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

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

Step 7.3.2.4.2.1.2

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

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

Step 7.3.2.4.2.2

Add $1$ and $2$ .

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

Step 7.3.3

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

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

Step 7.3.4

Move the negative in front of the fraction.

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

Step 7.4

Rewrite the left side as a result of differentiating a product.

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

Step 7.5

Set up an integral on each side.

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

Step 7.6

Integrate the left side.

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

Step 7.7

Integrate the right side.

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Step 7.7.1

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

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

Step 7.7.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 7.7.3

Simplify the expression.

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Step 7.7.3.1

Multiply $2$ by $- 1$ .

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

Step 7.7.3.2

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.7.3.3

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.7.3.3.1

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

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

Step 7.7.3.3.2

Multiply $2$ by $- 1$ .

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

Step 7.7.4

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

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

Step 7.7.5

Simplify the answer.

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Step 7.7.5.1

Rewrite $- 2 (- x^{- 1} + C)$ as $- 2 (- \frac{1}{x}) + C$ .

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

Step 7.7.5.2

Simplify.

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Step 7.7.5.2.1

Multiply $- 1$ by $- 2$ .

$\frac{1}{x^{2}} v = 2 \frac{1}{x} + C$

Step 7.7.5.2.2

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

$\frac{1}{x^{2}} v = \frac{2}{x} + C$

Step 7.8

Solve for $v$ .

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Step 7.8.1

Move all terms containing variables to the left side of the equation.

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Step 7.8.1.1

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

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

Step 7.8.1.2

Subtract $C$ from both sides of the equation.

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

Step 7.8.1.3

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

$\frac{v}{x^{2}} - \frac{2}{x} - C = 0$

Step 7.8.2

Move all terms not containing $v$ to the right side of the equation.

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Step 7.8.2.1

Add $\frac{2}{x}$ to both sides of the equation.

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

Step 7.8.2.2

Add $C$ to both sides of the equation.

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

Step 7.8.3

Multiply both sides by $x^{2}$ .

$\frac{v}{x^{2}} x^{2} = (\frac{2}{x} + C) x^{2}$

Step 7.8.4

Simplify.

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Step 7.8.4.1

Simplify the left side.

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Step 7.8.4.1.1

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

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Step 7.8.4.1.1.1

Cancel the common factor.

$\frac{v}{x^{2}} x^{2} = (\frac{2}{x} + C) x^{2}$

Step 7.8.4.1.1.2

Rewrite the expression.

$v = (\frac{2}{x} + C) x^{2}$

Step 7.8.4.2

Simplify the right side.

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Step 7.8.4.2.1

Simplify $(\frac{2}{x} + C) x^{2}$ .

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Step 7.8.4.2.1.1

Apply the distributive property.

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

Step 7.8.4.2.1.2

Cancel the common factor of $x$ .

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Step 7.8.4.2.1.2.1

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

$v = \frac{2}{x} (x \cdot x) + C x^{2}$

Step 7.8.4.2.1.2.2

Cancel the common factor.

$v = \frac{2}{x} (x \cdot x) + C x^{2}$

Step 7.8.4.2.1.2.3

Rewrite the expression.

$v = 2 x + C x^{2}$

Step 7.8.4.2.1.3

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

$v = C x^{2} + 2 x$

Step 8

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

$y^{- 2} = C x^{2} + 2 x$