Calculus Examples

$\frac{d y}{d x} = e^{2 x + 3 y}$ $y (0) = 0$

Step 1

Let $v = e^{2 x + 3 y}$ . Substitute $v$ for all occurrences of $e^{2 x + 3 y}$ .

$\frac{d y}{d x} = v$

Step 2

Find $\frac{d v}{d x}$ by differentiating $e^{2 x + 3 y}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x + 3 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $2 x + 3 y$ .

$\frac{d v}{d x} = \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [2 x + 3 y]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{d v}{d x} = e^{u_{1}} \frac{d}{d x} [2 x + 3 y]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $2 x + 3 y$ .

$\frac{d v}{d x} = e^{2 x + 3 y} \frac{d}{d x} [2 x + 3 y]$

Step 2.2

By the Sum Rule, the derivative of $2 x + 3 y$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3 y]$ .

$\frac{d v}{d x} = e^{2 x + 3 y} (\frac{d}{d x} [2 x] + \frac{d}{d x} [3 y])$

Step 2.3

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

$\frac{d v}{d x} = e^{2 x + 3 y} (2 \frac{d}{d x} [x] + \frac{d}{d x} [3 y])$

Step 2.4

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

$\frac{d v}{d x} = e^{2 x + 3 y} (2 \cdot 1 + \frac{d}{d x} [3 y])$

Step 2.5

Multiply $2$ by $1$ .

$\frac{d v}{d x} = e^{2 x + 3 y} (2 + \frac{d}{d x} [3 y])$

Step 2.6

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

$\frac{d v}{d x} = e^{2 x + 3 y} (2 + 3 \frac{d}{d x} [y])$

Step 2.7

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

$\frac{d v}{d x} = e^{2 x + 3 y} (2 + 3 y')$

Step 3

Substitute $v$ for $e^{2 x + 3 y}$ .

$\frac{d v}{d x} = v (2 + 3 y')$

Step 4

Substitute the derivative back in to the differential equation.

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

Step 5

Separate the variables.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Multiply both sides by $3 v$ .

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

Step 5.1.3

Simplify.

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Step 5.1.3.1

Simplify the left side.

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Step 5.1.3.1.1

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

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Step 5.1.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.3.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.3.1.1.2.1

Factor $3$ out of $3 v$ .

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

Step 5.1.3.1.1.2.2

Cancel the common factor.

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

Step 5.1.3.1.1.2.3

Rewrite the expression.

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

Step 5.1.3.1.1.3

Cancel the common factor of $v$ .

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Step 5.1.3.1.1.3.1

Cancel the common factor.

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

Step 5.1.3.1.1.3.2

Rewrite the expression.

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

Step 5.1.3.2

Simplify the right side.

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Step 5.1.3.2.1

Simplify $(v + \frac{2}{3}) (3 v)$ .

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Step 5.1.3.2.1.1

Apply the distributive property.

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

Step 5.1.3.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.3.2.1.3

Cancel the common factor of $3$ .

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Step 5.1.3.2.1.3.1

Cancel the common factor.

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

Step 5.1.3.2.1.3.2

Rewrite the expression.

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

Step 5.1.3.2.1.4

Multiply $v$ by $v$ by adding the exponents.

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Step 5.1.3.2.1.4.1

Move $v$ .

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

Step 5.1.3.2.1.4.2

Multiply $v$ by $v$ .

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

Step 5.2

Multiply both sides by $\frac{1}{3 v^{2} + 2 v}$ .

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

Step 5.3

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

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Rewrite the equation.

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

Step 6

Integrate both sides.

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Step 6.1

Set up an integral on each side.

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

Step 6.2

Integrate the left side.

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Step 6.2.1

Write the fraction using partial fraction decomposition.

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Step 6.2.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 6.2.1.1.1

Factor $v$ out of $3 v^{2} + 2 v$ .

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Step 6.2.1.1.1.1

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

$\frac{1}{v (3 v) + 2 v}$

Step 6.2.1.1.1.2

Factor $v$ out of $2 v$ .

$\frac{1}{v (3 v) + v \cdot 2}$

Step 6.2.1.1.1.3

Factor $v$ out of $v (3 v) + v \cdot 2$ .

$\frac{1}{v (3 v + 2)}$

Step 6.2.1.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{v} + \frac{B}{3 v + 2}$

Step 6.2.1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $v (3 v + 2)$ .

$\frac{1 (v (3 v + 2))}{v (3 v + 2)} = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.4

Cancel the common factor of $v$ .

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Step 6.2.1.1.4.1

Cancel the common factor.

$\frac{1 (v (3 v + 2))}{v (3 v + 2)} = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.4.2

Rewrite the expression.

$\frac{1 (3 v + 2)}{3 v + 2} = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.5

Cancel the common factor of $3 v + 2$ .

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Step 6.2.1.1.5.1

Cancel the common factor.

$\frac{1 (3 v + 2)}{3 v + 2} = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.5.2

Rewrite the expression.

$1 = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6

Simplify each term.

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Step 6.2.1.1.6.1

Cancel the common factor of $v$ .

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Step 6.2.1.1.6.1.1

Cancel the common factor.

$1 = \frac{A (v (3 v + 2))}{v} + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6.1.2

Divide $A (3 v + 2)$ by $1$ .

$1 = A (3 v + 2) + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6.2

Apply the distributive property.

$1 = A (3 v) + A \cdot 2 + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6.3

Rewrite using the commutative property of multiplication.

$1 = 3 A v + A \cdot 2 + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6.4

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

$1 = 3 A v + 2 A + \frac{B (v (3 v + 2))}{3 v + 2}$

Step 6.2.1.1.6.5

Divide $B v$ by $1$ .

$1 = 3 A v + 2 A + B v$

Step 6.2.1.1.7

Simplify the expression.

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Step 6.2.1.1.7.1

Move $A$ .

$1 = 3 v A + 2 A + B v$

Step 6.2.1.1.7.2

Reorder $B$ and $v$ .

$1 = 3 v A + 2 A + B v$

Step 6.2.1.1.7.3

Move $2 A$ .

$1 = 3 v A + B v + 2 A$

Step 6.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 6.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $v$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = 3 A + B$

Step 6.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $v$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 2 A$

Step 6.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = 3 A + B$

$1 = 2 A$

$0 = 3 A + B$

$1 = 2 A$

Step 6.2.1.3

Solve the system of equations.

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Step 6.2.1.3.1

Solve for $A$ in $1 = 2 A$ .

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Step 6.2.1.3.1.1

Rewrite the equation as $2 A = 1$ .

$2 A = 1$

$0 = 3 A + B$

Step 6.2.1.3.1.2

Divide each term in $2 A = 1$ by $2$ and simplify.

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Step 6.2.1.3.1.2.1

Divide each term in $2 A = 1$ by $2$ .

$\frac{2 A}{2} = \frac{1}{2}$

$0 = 3 A + B$

Step 6.2.1.3.1.2.2

Simplify the left side.

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Step 6.2.1.3.1.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.3.1.2.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{1}{2}$

$0 = 3 A + B$

Step 6.2.1.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{2}$

$0 = 3 A + B$

$A = \frac{1}{2}$

$0 = 3 A + B$

$A = \frac{1}{2}$

$0 = 3 A + B$

$A = \frac{1}{2}$

$0 = 3 A + B$

$A = \frac{1}{2}$

$0 = 3 A + B$

Step 6.2.1.3.2

Replace all occurrences of $A$ with $\frac{1}{2}$ in each equation.

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Step 6.2.1.3.2.1

Replace all occurrences of $A$ in $0 = 3 A + B$ with $\frac{1}{2}$ .

$0 = 3 (\frac{1}{2}) + B$

$A = \frac{1}{2}$

Step 6.2.1.3.2.2

Simplify the right side.

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Step 6.2.1.3.2.2.1

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

$0 = \frac{3}{2} + B$

$A = \frac{1}{2}$

$0 = \frac{3}{2} + B$

$A = \frac{1}{2}$

$0 = \frac{3}{2} + B$

$A = \frac{1}{2}$

Step 6.2.1.3.3

Solve for $B$ in $0 = \frac{3}{2} + B$ .

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Step 6.2.1.3.3.1

Rewrite the equation as $\frac{3}{2} + B = 0$ .

$\frac{3}{2} + B = 0$

$A = \frac{1}{2}$

Step 6.2.1.3.3.2

Subtract $\frac{3}{2}$ from both sides of the equation.

$B = - \frac{3}{2}$

$A = \frac{1}{2}$

$B = - \frac{3}{2}$

$A = \frac{1}{2}$

Step 6.2.1.3.4

Solve the system of equations.

$B = - \frac{3}{2} A = \frac{1}{2}$

Step 6.2.1.3.5

List all of the solutions.

$B = - \frac{3}{2}, A = \frac{1}{2}$

Step 6.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{v} + \frac{B}{3 v + 2}$ with the values found for $A$ and $B$ .

$\frac{\frac{1}{2}}{v} + \frac{- \frac{3}{2}}{3 v + 2}$

Step 6.2.1.5

Simplify.

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Step 6.2.1.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{1}{v} + \frac{- \frac{3}{2}}{3 v + 2}$

Step 6.2.1.5.2

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

$\frac{1}{2 v} + \frac{- \frac{3}{2}}{3 v + 2}$

Step 6.2.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2 v} - \frac{3}{2} \cdot \frac{1}{3 v + 2}$

Step 6.2.1.5.4

Multiply $\frac{1}{3 v + 2}$ by $\frac{3}{2}$ .

$\frac{1}{2 v} - \frac{3}{(3 v + 2) \cdot 2}$

Step 6.2.1.5.5

Move $2$ to the left of $3 v + 2$ .

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

Step 6.2.2

Split the single integral into multiple integrals.

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.2.6

Since $\frac{3}{2}$ is constant with respect to $v$ , move $\frac{3}{2}$ out of the integral.

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

Step 6.2.7

Let $u_{2} = 3 v + 2$ . Then $d u_{2} = 3 d v$ , so $\frac{1}{3} d u_{2} = d v$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.2.7.1

Let $u_{2} = 3 v + 2$ . Find $\frac{d u_{2}}{d v}$ .

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Step 6.2.7.1.1

Differentiate $3 v + 2$ .

$\frac{d}{d v} [3 v + 2]$

Step 6.2.7.1.2

By the Sum Rule, the derivative of $3 v + 2$ with respect to $v$ is $\frac{d}{d v} [3 v] + \frac{d}{d v} [2]$ .

$\frac{d}{d v} [3 v] + \frac{d}{d v} [2]$

Step 6.2.7.1.3

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

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Step 6.2.7.1.3.1

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

$3 \frac{d}{d v} [v] + \frac{d}{d v} [2]$

Step 6.2.7.1.3.2

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

$3 \cdot 1 + \frac{d}{d v} [2]$

Step 6.2.7.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d v} [2]$

Step 6.2.7.1.4

Differentiate using the Constant Rule.

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Step 6.2.7.1.4.1

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

$3 + 0$

Step 6.2.7.1.4.2

Add $3$ and $0$ .

$3$

Step 6.2.7.2

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

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{2} \int \frac{1}{u_{2}} \cdot \frac{1}{3} d u_{2} = \int d x$

Step 6.2.8

Simplify.

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Step 6.2.8.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{3}$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{2} \int \frac{1}{u_{2} \cdot 3} d u_{2} = \int d x$

Step 6.2.8.2

Move $3$ to the left of $u_{2}$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{2} \int \frac{1}{3 u_{2}} d u_{2} = \int d x$

Step 6.2.9

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{2} (\frac{1}{3} \int \frac{1}{u_{2}} d u_{2}) = \int d x$

Step 6.2.10

Simplify.

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Step 6.2.10.1

Multiply $\frac{1}{3}$ by $\frac{3}{2}$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{3 \cdot 2} \int \frac{1}{u_{2}} d u_{2} = \int d x$

Step 6.2.10.2

Multiply $3$ by $2$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3}{6} \int \frac{1}{u_{2}} d u_{2} = \int d x$

Step 6.2.10.3

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

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Step 6.2.10.3.1

Factor $3$ out of $3$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3 (1)}{6} \int \frac{1}{u_{2}} d u_{2} = \int d x$

Step 6.2.10.3.2

Cancel the common factors.

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Step 6.2.10.3.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3 \cdot 1}{3 \cdot 2} \int \frac{1}{u_{2}} d u_{2} = \int d x$

Step 6.2.10.3.2.2

Cancel the common factor.

$\frac{1}{2} (\ln (| v |) + C_{1}) - \frac{3 \cdot 1}{3 \cdot 2} \int \frac{1}{u_{2}} d u_{2} = \int d x$

Step 6.2.10.3.2.3

Rewrite the expression.

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

Step 6.2.11

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

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

Step 6.2.12

Simplify.

$\frac{1}{2} \ln (| v |) - \frac{1}{2} \ln (| u_{2} |) + C_{3} = \int d x$

Step 6.3

Apply the constant rule.

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

Step 6.4

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

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

Step 7

Solve for $v$ .

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

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

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

Step 7.1.1.2

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

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

Step 7.2

Multiply each term in $\frac{\ln (| v |)}{2} - \frac{\ln (| u_{2} |)}{2} = x + C$ by $2$ to eliminate the fractions.

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Step 7.2.1

Multiply each term in $\frac{\ln (| v |)}{2} - \frac{\ln (| u_{2} |)}{2} = x + C$ by $2$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1.1

Cancel the common factor.

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

Step 7.2.2.1.1.2

Rewrite the expression.

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

Step 7.2.2.1.2

Cancel the common factor of $2$ .

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Step 7.2.2.1.2.1

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

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

Step 7.2.2.1.2.2

Cancel the common factor.

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

Step 7.2.2.1.2.3

Rewrite the expression.

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

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

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

Step 7.2.3.1.2

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

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

Step 7.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 7.4

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

$e^{\ln (\frac{| v |}{| u_{2} |})} = e^{2 x + 2 C}$

Step 7.5

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

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

Step 7.6

Solve for $v$ .

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Step 7.6.1

Rewrite the equation as $\frac{| v |}{| u_{2} |} = e^{2 x + 2 C}$ .

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

Step 7.6.2

Multiply both sides by $| u_{2} |$ .

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

Step 7.6.3

Simplify the left side.

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Step 7.6.3.1

Cancel the common factor of $| u_{2} |$ .

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Step 7.6.3.1.1

Cancel the common factor.

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

Step 7.6.3.1.2

Rewrite the expression.

$| v | = e^{2 x + 2 C} | u_{2} |$

Step 7.6.4

Solve for $v$ .

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Step 7.6.4.1

Reorder factors in $e^{2 x + 2 C} | u_{2} |$ .

$| v | = | u_{2} | e^{2 x + 2 C}$

Step 7.6.4.2

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

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

Step 8

Group the constant terms together.

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Step 8.1

Simplify the constant of integration.

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

Step 8.2

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

$v = \pm | u_{2} | (e^{2 x} e^{C})$

Step 8.3

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

$v = \pm | u_{2} | (e^{C} e^{2 x})$

Step 8.4

Combine constants with the plus or minus.

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

Step 9

Replace all occurrences of $v$ with $e^{2 x + 3 y}$ .

$e^{2 x + 3 y} = C | u_{2} | e^{2 x}$

Step 10

Solve for $y$ .

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Step 10.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 x + 3 y}) = \ln (C | u_{2} | e^{2 x})$

Step 10.2

Expand the left side.

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Step 10.2.1

Expand $\ln (e^{2 x + 3 y})$ by moving $2 x + 3 y$ outside the logarithm.

$(2 x + 3 y) \ln (e) = \ln (C | u_{2} | e^{2 x})$

Step 10.2.2

The natural logarithm of $e$ is $1$ .

$(2 x + 3 y) \cdot 1 = \ln (C | u_{2} | e^{2 x})$

Step 10.2.3

Multiply $2 x + 3 y$ by $1$ .

$2 x + 3 y = \ln (C | u_{2} | e^{2 x})$

Step 10.3

Expand the right side.

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Step 10.3.1

Rewrite $\ln (C | u_{2} | e^{2 x})$ as $\ln (C | u_{2} |) + \ln (e^{2 x})$ .

$2 x + 3 y = \ln (C | u_{2} |) + \ln (e^{2 x})$

Step 10.3.2

Rewrite $\ln (C | u_{2} |)$ as $\ln (C) + \ln (| u_{2} |)$ .

$2 x + 3 y = \ln (C) + \ln (| u_{2} |) + \ln (e^{2 x})$

Step 10.3.3

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x + 3 y = \ln (C) + \ln (| u_{2} |) + 2 x \ln (e)$

Step 10.3.4

The natural logarithm of $e$ is $1$ .

$2 x + 3 y = \ln (C) + \ln (| u_{2} |) + 2 x \cdot 1$

Step 10.3.5

Multiply $2$ by $1$ .

$2 x + 3 y = \ln (C) + \ln (| u_{2} |) + 2 x$

Step 10.4

Simplify the right side.

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Step 10.4.1

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

$2 x + 3 y = \ln (C | u_{2} |) + 2 x$

Step 10.5

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

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Step 10.5.1

Subtract $2 x$ from both sides of the equation.

$3 y = \ln (C | u_{2} |) + 2 x - 2 x$

Step 10.5.2

Combine the opposite terms in $\ln (C | u_{2} |) + 2 x - 2 x$ .

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Step 10.5.2.1

Subtract $2 x$ from $2 x$ .

$3 y = \ln (C | u_{2} |) + 0$

Step 10.5.2.2

Add $\ln (C | u_{2} |)$ and $0$ .

$3 y = \ln (C | u_{2} |)$

Step 10.6

Divide each term in $3 y = \ln (C | u_{2} |)$ by $3$ and simplify.

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Step 10.6.1

Divide each term in $3 y = \ln (C | u_{2} |)$ by $3$ .

$\frac{3 y}{3} = \frac{\ln (C | u_{2} |)}{3}$

Step 10.6.2

Simplify the left side.

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Step 10.6.2.1

Cancel the common factor of $3$ .

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Step 10.6.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\ln (C | u_{2} |)}{3}$

Step 10.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (C | u_{2} |)}{3}$

Step 11

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \frac{\ln (C | u_{2} |)}{3}$ .

$0 = \frac{\ln (C | u_{2} |)}{3}$

Step 12

Solve for $C$ .

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Step 12.1

Set the numerator equal to zero.

$\ln (C | u_{2} |) = 0$

Step 12.2

Solve the equation for $C$ .

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Step 12.2.1

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

$e^{\ln (C | u_{2} |)} = e^{0}$

Step 12.2.2

Rewrite $\ln (C | u_{2} |) = 0$ 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 $b^{y} = x$ .

$e^{0} = C | u_{2} |$

Step 12.2.3

Solve for $C$ .

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Step 12.2.3.1

Rewrite the equation as $C | u_{2} | = e^{0}$ .

$C | u_{2} | = e^{0}$

Step 12.2.3.2

Anything raised to $0$ is $1$ .

$C | u_{2} | = 1$

Step 12.2.3.3

Divide each term in $C | u_{2} | = 1$ by $| u_{2} |$ and simplify.

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Step 12.2.3.3.1

Divide each term in $C | u_{2} | = 1$ by $| u_{2} |$ .

$\frac{C | u_{2} |}{| u_{2} |} = \frac{1}{| u_{2} |}$

Step 12.2.3.3.2

Simplify the left side.

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Step 12.2.3.3.2.1

Cancel the common factor of $| u_{2} |$ .

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Step 12.2.3.3.2.1.1

Cancel the common factor.

$\frac{C | u_{2} |}{| u_{2} |} = \frac{1}{| u_{2} |}$

Step 12.2.3.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{1}{| u_{2} |}$

Step 13

Substitute $\frac{1}{| u_{2} |}$ for $C$ in $y = \frac{\ln (C | u_{2} |)}{3}$ and simplify.

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Step 13.1

Substitute $\frac{1}{| u_{2} |}$ for $C$ .

$y = \frac{\ln (\frac{1}{| u_{2} |} | u_{2} |)}{3}$

Step 13.2

Rewrite $\frac{\ln (\frac{1}{| u_{2} |} | u_{2} |)}{3}$ as $\frac{1}{3} \ln (\frac{1}{| u_{2} |} | u_{2} |)$ .

$y = \frac{1}{3} \ln (\frac{1}{| u_{2} |} | u_{2} |)$

Step 13.3

Simplify $\frac{1}{3} \ln (\frac{1}{| u_{2} |} | u_{2} |)$ by moving $\frac{1}{3}$ inside the logarithm.

$y = \ln ({(\frac{1}{| u_{2} |} | u_{2} |)}^{\frac{1}{3}})$

Step 13.4

Cancel the common factor of $| u_{2} |$ .

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Step 13.4.1

Cancel the common factor.

$y = \ln ({(\frac{1}{| u_{2} |} | u_{2} |)}^{\frac{1}{3}})$

Step 13.4.2

Rewrite the expression.

$y = \ln (1^{\frac{1}{3}})$

Step 13.5

One to any power is one.

$y = \ln (1)$

Step 13.6

The natural logarithm of $1$ is $0$ .

$y = 0$