Calculus Examples

$(3 x - 1) \frac{d y}{d x} = 6 y - 10 {(3 x - 1)}^{\frac{1}{3}}$

Step 1

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

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Step 1.1

Subtract $6 y$ from both sides of the equation.

$(3 x - 1) \frac{d y}{d x} - 6 y = - 10 {(3 x - 1)}^{\frac{1}{3}}$

Step 1.2

Divide each term in $(3 x - 1) \frac{d y}{d x} - 6 y = - 10 {(3 x - 1)}^{\frac{1}{3}}$ by $3 x - 1$ .

$\frac{(3 x - 1) \frac{d y}{d x}}{3 x - 1} + \frac{- 6 y}{3 x - 1} = \frac{- 10 {(3 x - 1)}^{\frac{1}{3}}}{3 x - 1}$

Step 1.3

Cancel the common factor of $3 x - 1$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(3 x - 1) \frac{d y}{d x}}{3 x - 1} + \frac{- 6 y}{3 x - 1} = \frac{- 10 {(3 x - 1)}^{\frac{1}{3}}}{3 x - 1}$

Step 1.3.2

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

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = \frac{- 10 {(3 x - 1)}^{\frac{1}{3}}}{3 x - 1}$

Step 1.4

Factor $3 x - 1$ out of $- 10 {(3 x - 1)}^{\frac{1}{3}}$ .

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = \frac{(3 x - 1) (- 10 {(3 x - 1)}^{\frac{- 2}{3}})}{3 x - 1}$

Step 1.5

Cancel the common factors.

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Step 1.5.1

Multiply by $1$ .

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = \frac{(3 x - 1) (- 10 {(3 x - 1)}^{\frac{- 2}{3}})}{(3 x - 1) \cdot 1}$

Step 1.5.2

Cancel the common factor.

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = \frac{(3 x - 1) (- 10 {(3 x - 1)}^{\frac{- 2}{3}})}{(3 x - 1) \cdot 1}$

Step 1.5.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = \frac{- 10 {(3 x - 1)}^{\frac{- 2}{3}}}{1}$

Step 1.5.4

Divide $- 10 {(3 x - 1)}^{\frac{- 2}{3}}$ by $1$ .

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = - 10 {(3 x - 1)}^{\frac{- 2}{3}}$

Step 1.6

Move the negative in front of the fraction.

$\frac{d y}{d x} + \frac{- 6 y}{3 x - 1} = - 10 {(3 x - 1)}^{- \frac{2}{3}}$

Step 1.7

Factor $y$ out of $\frac{- 6 y}{3 x - 1}$ .

$\frac{d y}{d x} + y \frac{- 6}{3 x - 1} = - 10 {(3 x - 1)}^{- \frac{2}{3}}$

Step 1.8

Reorder $y$ and $\frac{- 6}{3 x - 1}$ .

$\frac{d y}{d x} + \frac{- 6}{3 x - 1} y = - 10 {(3 x - 1)}^{- \frac{2}{3}}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 6}{3 x - 1}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 6}{3 x - 1} d x}$

Step 2.2

Integrate $\frac{- 6}{3 x - 1}$ .

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Step 2.2.1

Move the negative in front of the fraction.

$e^{\int - \frac{6}{3 x - 1} d x}$

Step 2.2.2

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

$e^{- \int \frac{6}{3 x - 1} d x}$

Step 2.2.3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{- (6 \int \frac{1}{3 x - 1} d x)}$

Step 2.2.4

Multiply $6$ by $- 1$ .

$e^{- 6 \int \frac{1}{3 x - 1} d x}$

Step 2.2.5

Let $u = 3 x - 1$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.5.1

Let $u = 3 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.2.5.1.1

Differentiate $3 x - 1$ .

$\frac{d}{d x} [3 x - 1]$

Step 2.2.5.1.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$

Step 2.2.5.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.2.5.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 2.2.5.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [- 1]$

Step 2.2.5.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 1]$

Step 2.2.5.1.4

Differentiate using the Constant Rule.

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Step 2.2.5.1.4.1

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

$3 + 0$

Step 2.2.5.1.4.2

Add $3$ and $0$ .

$3$

Step 2.2.5.2

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

$e^{- 6 \int \frac{1}{u} \cdot \frac{1}{3} d u}$

Step 2.2.6

Simplify.

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Step 2.2.6.1

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

$e^{- 6 \int \frac{1}{u \cdot 3} d u}$

Step 2.2.6.2

Move $3$ to the left of $u$ .

$e^{- 6 \int \frac{1}{3 u} d u}$

Step 2.2.7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$e^{- 6 (\frac{1}{3} \int \frac{1}{u} d u)}$

Step 2.2.8

Simplify.

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Step 2.2.8.1

Combine $\frac{1}{3}$ and $- 6$ .

$e^{\frac{- 6}{3} \int \frac{1}{u} d u}$

Step 2.2.8.2

Cancel the common factor of $- 6$ and $3$ .

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Step 2.2.8.2.1

Factor $3$ out of $- 6$ .

$e^{\frac{3 \cdot - 2}{3} \int \frac{1}{u} d u}$

Step 2.2.8.2.2

Cancel the common factors.

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Step 2.2.8.2.2.1

Factor $3$ out of $3$ .

$e^{\frac{3 \cdot - 2}{3 (1)} \int \frac{1}{u} d u}$

Step 2.2.8.2.2.2

Cancel the common factor.

$e^{\frac{3 \cdot - 2}{3 \cdot 1} \int \frac{1}{u} d u}$

Step 2.2.8.2.2.3

Rewrite the expression.

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

Step 2.2.8.2.2.4

Divide $- 2$ by $1$ .

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

Step 2.2.9

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

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

Step 2.2.10

Simplify.

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

Step 2.2.11

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 2.3

Remove the constant of integration.

$e^{- 2 \ln (3 x - 1)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln ({(3 x - 1)}^{- 2})}$

Step 2.5

Exponentiation and log are inverse functions.

${(3 x - 1)}^{- 2}$

Step 2.6

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

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

Step 3

Multiply each term by the integrating factor $\frac{1}{{(3 x - 1)}^{2}}$ .

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Step 3.1

Multiply each term by $\frac{1}{{(3 x - 1)}^{2}}$ .

$\frac{1}{{(3 x - 1)}^{2}} \frac{d y}{d x} + \frac{1}{{(3 x - 1)}^{2}} (\frac{- 6}{3 x - 1} y) = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{{(3 x - 1)}^{2}}$ and $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} + \frac{1}{{(3 x - 1)}^{2}} (\frac{- 6}{3 x - 1} y) = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} + \frac{1}{{(3 x - 1)}^{2}} (- \frac{6}{3 x - 1} y) = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{1}{{(3 x - 1)}^{2}} (\frac{6}{3 x - 1} y) = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.4

Combine $\frac{6}{3 x - 1}$ and $y$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{1}{{(3 x - 1)}^{2}} \cdot \frac{6 y}{3 x - 1} = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.5

Multiply $- \frac{1}{{(3 x - 1)}^{2}} \cdot \frac{6 y}{3 x - 1}$ .

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Step 3.2.5.1

Multiply $\frac{6 y}{3 x - 1}$ by $\frac{1}{{(3 x - 1)}^{2}}$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{(3 x - 1) {(3 x - 1)}^{2}} = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.5.2

Multiply $3 x - 1$ by ${(3 x - 1)}^{2}$ by adding the exponents.

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Step 3.2.5.2.1

Multiply $3 x - 1$ by ${(3 x - 1)}^{2}$ .

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Step 3.2.5.2.1.1

Raise $3 x - 1$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{1} {(3 x - 1)}^{2}} = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.5.2.1.2

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

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{1 + 2}} = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.2.5.2.2

Add $1$ and $2$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = \frac{1}{{(3 x - 1)}^{2}} (- 10 {(3 x - 1)}^{- \frac{2}{3}})$

Step 3.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = - 10 \frac{1}{{(3 x - 1)}^{2}} {(3 x - 1)}^{- \frac{2}{3}}$

Step 3.4

Combine $- 10$ and $\frac{1}{{(3 x - 1)}^{2}}$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = \frac{- 10}{{(3 x - 1)}^{2}} {(3 x - 1)}^{- \frac{2}{3}}$

Step 3.5

Cancel the common factor of ${(3 x - 1)}^{- \frac{2}{3}}$ .

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Step 3.5.1

Factor ${(3 x - 1)}^{- \frac{2}{3}}$ out of ${(3 x - 1)}^{2}$ .

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = \frac{- 10}{{(3 x - 1)}^{- \frac{2}{3}} {(3 x - 1)}^{\frac{8}{3}}} {(3 x - 1)}^{- \frac{2}{3}}$

Step 3.5.2

Cancel the common factor.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = \frac{- 10}{{(3 x - 1)}^{- \frac{2}{3}} {(3 x - 1)}^{\frac{8}{3}}} {(3 x - 1)}^{- \frac{2}{3}}$

Step 3.5.3

Rewrite the expression.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = \frac{- 10}{{(3 x - 1)}^{\frac{8}{3}}}$

Step 3.6

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{{(3 x - 1)}^{2}} - \frac{6 y}{{(3 x - 1)}^{3}} = - \frac{10}{{(3 x - 1)}^{\frac{8}{3}}}$

Step 4

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

$\frac{d}{d x} [\frac{1}{{(3 x - 1)}^{2}} y] = - \frac{10}{{(3 x - 1)}^{\frac{8}{3}}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{{(3 x - 1)}^{2}} y] d x = \int - \frac{10}{{(3 x - 1)}^{\frac{8}{3}}} d x$

Step 6

Integrate the left side.

$\frac{1}{{(3 x - 1)}^{2}} y = \int - \frac{10}{{(3 x - 1)}^{\frac{8}{3}}} d x$

Step 7

Integrate the right side.

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Step 7.1

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

$\frac{1}{{(3 x - 1)}^{2}} y = - \int \frac{10}{{(3 x - 1)}^{\frac{8}{3}}} d x$

Step 7.2

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$\frac{1}{{(3 x - 1)}^{2}} y = - (10 \int \frac{1}{{(3 x - 1)}^{\frac{8}{3}}} d x)$

Step 7.3

Multiply $10$ by $- 1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - 10 \int \frac{1}{{(3 x - 1)}^{\frac{8}{3}}} d x$

Step 7.4

Let $u = 3 x - 1$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.4.1

Let $u = 3 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 7.4.1.1

Differentiate $3 x - 1$ .

$\frac{d}{d x} [3 x - 1]$

Step 7.4.1.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$

Step 7.4.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 7.4.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 7.4.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [- 1]$

Step 7.4.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 1]$

Step 7.4.1.4

Differentiate using the Constant Rule.

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Step 7.4.1.4.1

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

$3 + 0$

Step 7.4.1.4.2

Add $3$ and $0$ .

$3$

Step 7.4.2

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

$\frac{1}{{(3 x - 1)}^{2}} y = - 10 \int \frac{1}{u^{\frac{8}{3}}} \cdot \frac{1}{3} d u$

Step 7.5

Simplify.

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Step 7.5.1

Multiply $\frac{1}{u^{\frac{8}{3}}}$ by $\frac{1}{3}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - 10 \int \frac{1}{u^{\frac{8}{3}} \cdot 3} d u$

Step 7.5.2

Move $3$ to the left of $u^{\frac{8}{3}}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - 10 \int \frac{1}{3 u^{\frac{8}{3}}} d u$

Step 7.6

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{{(3 x - 1)}^{2}} y = - 10 (\frac{1}{3} \int \frac{1}{u^{\frac{8}{3}}} d u)$

Step 7.7

Simplify the expression.

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Step 7.7.1

Simplify.

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Step 7.7.1.1

Combine $\frac{1}{3}$ and $- 10$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{- 10}{3} \int \frac{1}{u^{\frac{8}{3}}} d u$

Step 7.7.1.2

Move the negative in front of the fraction.

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int \frac{1}{u^{\frac{8}{3}}} d u$

Step 7.7.2

Apply basic rules of exponents.

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Step 7.7.2.1

Move $u^{\frac{8}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int {(u^{\frac{8}{3}})}^{- 1} d u$

Step 7.7.2.2

Multiply the exponents in ${(u^{\frac{8}{3}})}^{- 1}$ .

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Step 7.7.2.2.1

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

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int u^{\frac{8}{3} \cdot - 1} d u$

Step 7.7.2.2.2

Multiply $\frac{8}{3} \cdot - 1$ .

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Step 7.7.2.2.2.1

Combine $\frac{8}{3}$ and $- 1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int u^{\frac{8 \cdot - 1}{3}} d u$

Step 7.7.2.2.2.2

Multiply $8$ by $- 1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int u^{\frac{- 8}{3}} d u$

Step 7.7.2.2.3

Move the negative in front of the fraction.

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} \int u^{- \frac{8}{3}} d u$

Step 7.8

By the Power Rule, the integral of $u^{- \frac{8}{3}}$ with respect to $u$ is $- \frac{3}{5} u^{- \frac{5}{3}}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} (- \frac{3}{5} u^{- \frac{5}{3}} + C)$

Step 7.9

Simplify.

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Step 7.9.1

Rewrite $- \frac{10}{3} (- \frac{3}{5} u^{- \frac{5}{3}} + C)$ as $- \frac{10}{3} (- \frac{3}{5} \cdot \frac{1}{u^{\frac{5}{3}}}) + C$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} (- \frac{3}{5} \cdot \frac{1}{u^{\frac{5}{3}}}) + C$

Step 7.9.2

Simplify.

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Step 7.9.2.1

Multiply $\frac{1}{u^{\frac{5}{3}}}$ by $\frac{3}{5}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} (- \frac{3}{u^{\frac{5}{3}} \cdot 5}) + C$

Step 7.9.2.2

Move $5$ to the left of $u^{\frac{5}{3}}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = - \frac{10}{3} (- \frac{3}{5 \cdot u^{\frac{5}{3}}}) + C$

Step 7.9.2.3

Multiply $- 1$ by $- 1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = 1 (\frac{10}{3}) \frac{3}{5 u^{\frac{5}{3}}} + C$

Step 7.9.2.4

Multiply $\frac{10}{3}$ by $1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{10}{3} \cdot \frac{3}{5 u^{\frac{5}{3}}} + C$

Step 7.9.2.5

Multiply $\frac{10}{3}$ by $\frac{3}{5 u^{\frac{5}{3}}}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{10 \cdot 3}{3 (5 u^{\frac{5}{3}})} + C$

Step 7.9.2.6

Multiply $10$ by $3$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{30}{3 (5 u^{\frac{5}{3}})} + C$

Step 7.9.2.7

Multiply $5$ by $3$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{30}{15 u^{\frac{5}{3}}} + C$

Step 7.9.2.8

Factor $15$ out of $30$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{15 \cdot 2}{15 u^{\frac{5}{3}}} + C$

Step 7.9.2.9

Cancel the common factors.

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Step 7.9.2.9.1

Factor $15$ out of $15 u^{\frac{5}{3}}$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{15 \cdot 2}{15 (u^{\frac{5}{3}})} + C$

Step 7.9.2.9.2

Cancel the common factor.

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{15 \cdot 2}{15 u^{\frac{5}{3}}} + C$

Step 7.9.2.9.3

Rewrite the expression.

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{2}{u^{\frac{5}{3}}} + C$

Step 7.10

Replace all occurrences of $u$ with $3 x - 1$ .

$\frac{1}{{(3 x - 1)}^{2}} y = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C$

Step 8

Solve for $y$ .

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Step 8.1

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

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Step 8.1.1

Subtract $\frac{2}{{(3 x - 1)}^{\frac{5}{3}}}$ from both sides of the equation.

$\frac{1}{{(3 x - 1)}^{2}} y - \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} = C$

Step 8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{{(3 x - 1)}^{2}} y - \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} - C = 0$

Step 8.1.3

Combine $\frac{1}{{(3 x - 1)}^{2}}$ and $y$ .

$\frac{y}{{(3 x - 1)}^{2}} - \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} - C = 0$

Step 8.2

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

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Step 8.2.1

Add $\frac{2}{{(3 x - 1)}^{\frac{5}{3}}}$ to both sides of the equation.

$\frac{y}{{(3 x - 1)}^{2}} - C = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}}$

Step 8.2.2

Add $C$ to both sides of the equation.

$\frac{y}{{(3 x - 1)}^{2}} = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C$

Step 8.3

Multiply both sides by ${(3 x - 1)}^{2}$ .

$\frac{y}{{(3 x - 1)}^{2}} {(3 x - 1)}^{2} = (\frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C) {(3 x - 1)}^{2}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Cancel the common factor of ${(3 x - 1)}^{2}$ .

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{{(3 x - 1)}^{2}} {(3 x - 1)}^{2} = (\frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C) {(3 x - 1)}^{2}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C) {(3 x - 1)}^{2}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{2}{{(3 x - 1)}^{\frac{5}{3}}} + C) {(3 x - 1)}^{2}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} {(3 x - 1)}^{2} + C {(3 x - 1)}^{2}$

Step 8.4.2.1.2

Cancel the common factor of ${(3 x - 1)}^{\frac{5}{3}}$ .

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Step 8.4.2.1.2.1

Factor ${(3 x - 1)}^{\frac{5}{3}}$ out of ${(3 x - 1)}^{2}$ .

$y = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} ({(3 x - 1)}^{\frac{5}{3}} {(3 x - 1)}^{\frac{1}{3}}) + C {(3 x - 1)}^{2}$

Step 8.4.2.1.2.2

Cancel the common factor.

$y = \frac{2}{{(3 x - 1)}^{\frac{5}{3}}} ({(3 x - 1)}^{\frac{5}{3}} {(3 x - 1)}^{\frac{1}{3}}) + C {(3 x - 1)}^{2}$

Step 8.4.2.1.2.3

Rewrite the expression.

$y = 2 {(3 x - 1)}^{\frac{1}{3}} + C {(3 x - 1)}^{2}$

Step 8.4.2.1.3

Reorder $2 {(3 x - 1)}^{\frac{1}{3}}$ and $C {(3 x - 1)}^{2}$ .

$y = C {(3 x - 1)}^{2} + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5

Simplify each term.

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Step 8.5.1

Rewrite ${(3 x - 1)}^{2}$ as $(3 x - 1) (3 x - 1)$ .

$y = C ((3 x - 1) (3 x - 1)) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.2

Expand $(3 x - 1) (3 x - 1)$ using the FOIL Method.

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Step 8.5.2.1

Apply the distributive property.

$y = C (3 x (3 x - 1) - 1 (3 x - 1)) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.2.2

Apply the distributive property.

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

Step 8.5.2.3

Apply the distributive property.

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

Step 8.5.3

Simplify and combine like terms.

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Step 8.5.3.1

Simplify each term.

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Step 8.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 8.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 8.5.3.1.2.1

Move $x$ .

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

Step 8.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 8.5.3.1.3

Multiply $3$ by $3$ .

$y = C (9 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.3.1.4

Multiply $- 1$ by $3$ .

$y = C (9 x^{2} - 3 x - 1 (3 x) - 1 \cdot - 1) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.3.1.5

Multiply $3$ by $- 1$ .

$y = C (9 x^{2} - 3 x - 3 x - 1 \cdot - 1) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.3.1.6

Multiply $- 1$ by $- 1$ .

$y = C (9 x^{2} - 3 x - 3 x + 1) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.3.2

Subtract $3 x$ from $- 3 x$ .

$y = C (9 x^{2} - 6 x + 1) + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.4

Apply the distributive property.

$y = C (9 x^{2}) + C (- 6 x) + C \cdot 1 + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.5

Simplify.

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Step 8.5.5.1

Rewrite using the commutative property of multiplication.

$y = 9 C x^{2} + C (- 6 x) + C \cdot 1 + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.5.2

Rewrite using the commutative property of multiplication.

$y = 9 C x^{2} - 6 C x + C \cdot 1 + 2 {(3 x - 1)}^{\frac{1}{3}}$

Step 8.5.5.3

Multiply $C$ by $1$ .

$y = 9 C x^{2} - 6 C x + C + 2 {(3 x - 1)}^{\frac{1}{3}}$