Calculus Examples

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

Step 1

Let $v = y^{2}$ . Substitute $v$ for all occurrences of $y^{2}$ .

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

Step 2

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

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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 $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.1.1

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

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

Step 2.1.2

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

$\frac{d v}{d x} = 2 u \frac{d}{d x} [y]$

Step 2.1.3

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

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

Step 2.2

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

$\frac{d v}{d x} = 2 y y'$

Step 3

Substitute the derivative back in to the differential equation.

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Add $x$ to both sides of the equation.

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

Step 4.2

Multiply each term in $\frac{1}{2} \frac{d v}{d x} - 2 v = x$ by $2$ .

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Multiply $2$ by $- 2$ .

$\frac{d v}{d x} - 4 v = x \cdot 2$

Step 4.6

Reorder $x$ and $2$ .

$\frac{d v}{d x} - 4 v = 2 \cdot x$

Step 5

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 4$ .

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Step 5.1

Set up the integration.

$e^{\int - 4 d x}$

Step 5.2

Apply the constant rule.

$e^{- 4 x + C}$

Step 5.3

Remove the constant of integration.

$e^{- 4 x}$

Step 6

Multiply each term by the integrating factor $e^{- 4 x}$ .

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Step 6.1

Multiply each term by $e^{- 4 x}$ .

$e^{- 4 x} \frac{d v}{d x} + e^{- 4 x} (- 4 v) = e^{- 4 x} (2 \cdot x)$

Step 6.2

Rewrite using the commutative property of multiplication.

$e^{- 4 x} \frac{d v}{d x} - 4 e^{- 4 x} v = e^{- 4 x} (2 \cdot x)$

Step 6.3

Rewrite using the commutative property of multiplication.

$e^{- 4 x} \frac{d v}{d x} - 4 e^{- 4 x} v = 2 e^{- 4 x} x$

Step 6.4

Reorder factors in $e^{- 4 x} \frac{d v}{d x} - 4 e^{- 4 x} v = 2 e^{- 4 x} x$ .

$e^{- 4 x} \frac{d v}{d x} - 4 v e^{- 4 x} = 2 x e^{- 4 x}$

Step 7

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

$\frac{d}{d x} [e^{- 4 x} v] = 2 x e^{- 4 x}$

Step 8

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- 4 x} v] d x = \int 2 x e^{- 4 x} d x$

Step 9

Integrate the left side.

$e^{- 4 x} v = \int 2 x e^{- 4 x} d x$

Step 10

Integrate the right side.

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Step 10.1

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

$e^{- 4 x} v = 2 \int x e^{- 4 x} d x$

Step 10.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 4 x}$ .

$e^{- 4 x} v = 2 (x (- \frac{1}{4} e^{- 4 x}) - \int - \frac{1}{4} e^{- 4 x} d x)$

Step 10.3

Simplify.

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Step 10.3.1

Combine $e^{- 4 x}$ and $\frac{1}{4}$ .

$e^{- 4 x} v = 2 (x (- \frac{e^{- 4 x}}{4}) - \int - \frac{1}{4} e^{- 4 x} d x)$

Step 10.3.2

Combine $x$ and $\frac{e^{- 4 x}}{4}$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - \int - \frac{1}{4} e^{- 4 x} d x)$

Step 10.3.3

Combine $e^{- 4 x}$ and $\frac{1}{4}$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - \int - \frac{e^{- 4 x}}{4} d x)$

Step 10.4

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - - \int \frac{e^{- 4 x}}{4} d x)$

Step 10.5

Simplify.

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Step 10.5.1

Multiply $- 1$ by $- 1$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + 1 \int \frac{e^{- 4 x}}{4} d x)$

Step 10.5.2

Multiply $\int \frac{e^{- 4 x}}{4} d x$ by $1$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \int \frac{e^{- 4 x}}{4} d x)$

Step 10.6

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} \int e^{- 4 x} d x)$

Step 10.7

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

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Step 10.7.1

Let $u = - 4 x$ . Find $\frac{d u}{d x}$ .

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Step 10.7.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 10.7.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 10.7.1.3

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

$- 4 \cdot 1$

Step 10.7.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 10.7.2

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} \int e^{u} \frac{1}{- 4} d u)$

Step 10.8

Simplify.

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Step 10.8.1

Move the negative in front of the fraction.

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} \int e^{u} (- \frac{1}{4}) d u)$

Step 10.8.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} \int - \frac{e^{u}}{4} d u)$

Step 10.9

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} (- \int \frac{e^{u}}{4} d u))$

Step 10.10

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} + \frac{1}{4} (- (\frac{1}{4} \int e^{u} d u)))$

Step 10.11

Simplify.

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Step 10.11.1

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

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - \frac{1}{4 \cdot 4} \int e^{u} d u)$

Step 10.11.2

Multiply $4$ by $4$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - \frac{1}{16} \int e^{u} d u)$

Step 10.12

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{- 4 x} v = 2 (- \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u} + C))$

Step 10.13

Simplify.

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Step 10.13.1

Rewrite $2 (- \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u} + C))$ as $2 (- \frac{1}{4} x e^{- 4 x} - \frac{1}{16} e^{u}) + C$ .

$e^{- 4 x} v = 2 (- \frac{1}{4} x e^{- 4 x} - \frac{1}{16} e^{u}) + C$

Step 10.13.2

Simplify.

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Step 10.13.2.1

Combine $x$ and $\frac{1}{4}$ .

$e^{- 4 x} v = 2 (- \frac{x}{4} e^{- 4 x} - \frac{1}{16} e^{u}) + C$

Step 10.13.2.2

Combine $e^{- 4 x}$ and $\frac{x}{4}$ .

$e^{- 4 x} v = 2 (- \frac{e^{- 4 x} x}{4} - \frac{1}{16} e^{u}) + C$

Step 10.13.2.3

Combine $e^{u}$ and $\frac{1}{16}$ .

$e^{- 4 x} v = 2 (- \frac{e^{- 4 x} x}{4} - \frac{e^{u}}{16}) + C$

Step 10.14

Replace all occurrences of $u$ with $- 4 x$ .

$e^{- 4 x} v = 2 (- \frac{e^{- 4 x} x}{4} - \frac{e^{- 4 x}}{16}) + C$

Step 10.15

Simplify.

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Step 10.15.1

Apply the distributive property.

$e^{- 4 x} v = 2 (- \frac{e^{- 4 x} x}{4}) + 2 (- \frac{e^{- 4 x}}{16}) + C$

Step 10.15.2

Cancel the common factor of $2$ .

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Step 10.15.2.1

Move the leading negative in $- \frac{e^{- 4 x} x}{4}$ into the numerator.

$e^{- 4 x} v = 2 \frac{- e^{- 4 x} x}{4} + 2 (- \frac{e^{- 4 x}}{16}) + C$

Step 10.15.2.2

Factor $2$ out of $4$ .

$e^{- 4 x} v = 2 \frac{- e^{- 4 x} x}{2 (2)} + 2 (- \frac{e^{- 4 x}}{16}) + C$

Step 10.15.2.3

Cancel the common factor.

$e^{- 4 x} v = 2 \frac{- e^{- 4 x} x}{2 \cdot 2} + 2 (- \frac{e^{- 4 x}}{16}) + C$

Step 10.15.2.4

Rewrite the expression.

$e^{- 4 x} v = \frac{- e^{- 4 x} x}{2} + 2 (- \frac{e^{- 4 x}}{16}) + C$

Step 10.15.3

Cancel the common factor of $2$ .

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Step 10.15.3.1

Move the leading negative in $- \frac{e^{- 4 x}}{16}$ into the numerator.

$e^{- 4 x} v = \frac{- e^{- 4 x} x}{2} + 2 \frac{- e^{- 4 x}}{16} + C$

Step 10.15.3.2

Factor $2$ out of $16$ .

$e^{- 4 x} v = \frac{- e^{- 4 x} x}{2} + 2 \frac{- e^{- 4 x}}{2 (8)} + C$

Step 10.15.3.3

Cancel the common factor.

$e^{- 4 x} v = \frac{- e^{- 4 x} x}{2} + 2 \frac{- e^{- 4 x}}{2 \cdot 8} + C$

Step 10.15.3.4

Rewrite the expression.

$e^{- 4 x} v = \frac{- e^{- 4 x} x}{2} + \frac{- e^{- 4 x}}{8} + C$

Step 10.15.4

Simplify each term.

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Step 10.15.4.1

Move the negative in front of the fraction.

$e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} + \frac{- e^{- 4 x}}{8} + C$

Step 10.15.4.2

Move the negative in front of the fraction.

$e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8} + C$

Step 10.15.5

Reorder factors in $- \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8}$ .

$e^{- 4 x} v = - \frac{x e^{- 4 x}}{2} - \frac{e^{- 4 x}}{8} + C$

Step 10.16

Reorder terms.

$e^{- 4 x} v = - \frac{1}{2} x e^{- 4 x} - \frac{1}{8} e^{- 4 x} + C$

Step 11

Solve for $v$ .

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Step 11.1

Simplify.

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Step 11.1.1

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

$e^{- 4 x} v = - \frac{x}{2} e^{- 4 x} - \frac{1}{8} e^{- 4 x} + C$

Step 11.1.2

Combine $e^{- 4 x}$ and $\frac{x}{2}$ .

$e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} - \frac{1}{8} e^{- 4 x} + C$

Step 11.1.3

Combine $e^{- 4 x}$ and $\frac{1}{8}$ .

$e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8} + C$

Step 11.2

Divide each term in $e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8} + C$ by $e^{- 4 x}$ and simplify.

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Step 11.2.1

Divide each term in $e^{- 4 x} v = - \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8} + C$ by $e^{- 4 x}$ .

$\frac{e^{- 4 x} v}{e^{- 4 x}} = \frac{- \frac{e^{- 4 x} x}{2}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

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

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{e^{- 4 x} v}{e^{- 4 x}} = \frac{- \frac{e^{- 4 x} x}{2}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.2.1.2

Divide $v$ by $1$ .

$v = \frac{- \frac{e^{- 4 x} x}{2}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Simplify each term.

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Step 11.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$v = - \frac{e^{- 4 x} x}{2} \cdot \frac{1}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.2

Multiply $\frac{1}{e^{- 4 x}}$ by $\frac{e^{- 4 x} x}{2}$ .

$v = - \frac{e^{- 4 x} x}{e^{- 4 x} \cdot 2} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.3

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

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Step 11.2.3.1.3.1

Cancel the common factor.

$v = - \frac{e^{- 4 x} x}{e^{- 4 x} \cdot 2} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.3.2

Rewrite the expression.

$v = - \frac{x}{2} + \frac{- \frac{e^{- 4 x}}{8}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.4

Multiply the numerator by the reciprocal of the denominator.

$v = - \frac{x}{2} - \frac{e^{- 4 x}}{8} \cdot \frac{1}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.5

Multiply $\frac{1}{e^{- 4 x}}$ by $\frac{e^{- 4 x}}{8}$ .

$v = - \frac{x}{2} - \frac{e^{- 4 x}}{e^{- 4 x} \cdot 8} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.6

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

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Step 11.2.3.1.6.1

Cancel the common factor.

$v = - \frac{x}{2} - \frac{e^{- 4 x}}{e^{- 4 x} \cdot 8} + \frac{C}{e^{- 4 x}}$

Step 11.2.3.1.6.2

Rewrite the expression.

$v = - \frac{x}{2} - \frac{1}{8} + \frac{C}{e^{- 4 x}}$

Step 12

Replace all occurrences of $v$ with $y^{2}$ .

$y^{2} = - \frac{x}{2} - \frac{1}{8} + \frac{C}{e^{- 4 x}}$

Step 13

Solve for $y$ .

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Step 13.1

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

$y = \pm \sqrt{- \frac{x}{2} - \frac{1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2

Simplify $\pm \sqrt{- \frac{x}{2} - \frac{1}{8} + \frac{C}{e^{- 4 x}}}$ .

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Step 13.2.1

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{- \frac{x}{2} \cdot \frac{4}{4} - \frac{1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.2.2.1

Multiply $\frac{x}{2}$ by $\frac{4}{4}$ .

$y = \pm \sqrt{- \frac{x \cdot 4}{2 \cdot 4} - \frac{1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2.2.2

Multiply $2$ by $4$ .

$y = \pm \sqrt{- \frac{x \cdot 4}{8} - \frac{1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- x \cdot 4 - 1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2.4

Multiply $4$ by $- 1$ .

$y = \pm \sqrt{\frac{- 4 x - 1}{8} + \frac{C}{e^{- 4 x}}}$

Step 13.2.5

To write $\frac{- 4 x - 1}{8}$ as a fraction with a common denominator, multiply by $\frac{e^{- 4 x}}{e^{- 4 x}}$ .

$y = \pm \sqrt{\frac{- 4 x - 1}{8} \cdot \frac{e^{- 4 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}}$

Step 13.2.6

To write $\frac{C}{e^{- 4 x}}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \pm \sqrt{\frac{- 4 x - 1}{8} \cdot \frac{e^{- 4 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}} \cdot \frac{8}{8}}$

Step 13.2.7

Write each expression with a common denominator of $8 e^{- 4 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.2.7.1

Multiply $\frac{- 4 x - 1}{8}$ by $\frac{e^{- 4 x}}{e^{- 4 x}}$ .

$y = \pm \sqrt{\frac{(- 4 x - 1) e^{- 4 x}}{8 e^{- 4 x}} + \frac{C}{e^{- 4 x}} \cdot \frac{8}{8}}$

Step 13.2.7.2

Multiply $\frac{C}{e^{- 4 x}}$ by $\frac{8}{8}$ .

$y = \pm \sqrt{\frac{(- 4 x - 1) e^{- 4 x}}{8 e^{- 4 x}} + \frac{C \cdot 8}{e^{- 4 x} \cdot 8}}$

Step 13.2.7.3

Reorder the factors of $e^{- 4 x} \cdot 8$ .

$y = \pm \sqrt{\frac{(- 4 x - 1) e^{- 4 x}}{8 e^{- 4 x}} + \frac{C \cdot 8}{8 e^{- 4 x}}}$

Step 13.2.8

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{(- 4 x - 1) e^{- 4 x} + C \cdot 8}{8 e^{- 4 x}}}$

Step 13.2.9

Simplify the numerator.

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Step 13.2.9.1

Apply the distributive property.

$y = \pm \sqrt{\frac{- 4 x e^{- 4 x} - 1 e^{- 4 x} + C \cdot 8}{8 e^{- 4 x}}}$

Step 13.2.9.2

Rewrite $- 1 e^{- 4 x}$ as $- e^{- 4 x}$ .

$y = \pm \sqrt{\frac{- 4 x e^{- 4 x} - e^{- 4 x} + C \cdot 8}{8 e^{- 4 x}}}$

Step 13.2.9.3

Move $8$ to the left of $C$ .

$y = \pm \sqrt{\frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{8 e^{- 4 x}}}$

Step 13.2.10

Rewrite $\frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{8 e^{- 4 x}}$ as ${(\frac{1}{2 e^{- 2 x}})}^{2} \frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{2}$ .

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Step 13.2.10.1

Factor the perfect power $1^{2}$ out of $- 4 x e^{- 4 x} - e^{- 4 x} + 8 C$ .

$y = \pm \sqrt{\frac{1^{2} (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}{8 e^{- 4 x}}}$

Step 13.2.10.2

Factor the perfect power ${(2 e^{- 2 x})}^{2}$ out of $8 e^{- 4 x}$ .

$y = \pm \sqrt{\frac{1^{2} (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}{{(2 e^{- 2 x})}^{2} \cdot 2}}$

Step 13.2.10.3

Rearrange the fraction $\frac{1^{2} (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}{{(2 e^{- 2 x})}^{2} \cdot 2}$ .

$y = \pm \sqrt{{(\frac{1}{2 e^{- 2 x}})}^{2} \frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{2}}$

Step 13.2.11

Pull terms out from under the radical.

$y = \pm \frac{1}{2 e^{- 2 x}} \sqrt{\frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{2}}$

Step 13.2.12

Rewrite $\sqrt{\frac{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}{2}}$ as $\frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{\sqrt{2}}$ .

$y = \pm \frac{1}{2 e^{- 2 x}} \cdot \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{\sqrt{2}}$

Step 13.2.13

Combine.

$y = \pm \frac{1 \sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{2 e^{- 2 x} \sqrt{2}}$

Step 13.2.14

Multiply $\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}$ by $1$ .

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{2 e^{- 2 x} \sqrt{2}}$

Step 13.2.15

Multiply $\frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{2 e^{- 2 x} \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{2 e^{- 2 x} \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 13.2.16

Combine and simplify the denominator.

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Step 13.2.16.1

Multiply $\frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C}}{2 e^{- 2 x} \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \sqrt{2} \sqrt{2}}$

Step 13.2.16.2

Move $\sqrt{2}$ .

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} (\sqrt{2} \sqrt{2})}$

Step 13.2.16.3

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} ({\sqrt{2}}^{1} \sqrt{2})}$

Step 13.2.16.4

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 13.2.16.5

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} {\sqrt{2}}^{1 + 1}}$

Step 13.2.16.6

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} {\sqrt{2}}^{2}}$

Step 13.2.16.7

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

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Step 13.2.16.7.1

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} {(2^{\frac{1}{2}})}^{2}}$

Step 13.2.16.7.2

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 13.2.16.7.3

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

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \cdot 2^{\frac{2}{2}}}$

Step 13.2.16.7.4

Cancel the common factor of $2$ .

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Step 13.2.16.7.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \cdot 2^{\frac{2}{2}}}$

Step 13.2.16.7.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \cdot 2^{1}}$

Step 13.2.16.7.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{- 4 x e^{- 4 x} - e^{- 4 x} + 8 C} \sqrt{2}}{2 e^{- 2 x} \cdot 2}$

Step 13.2.17

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(- 4 x e^{- 4 x} - e^{- 4 x} + 8 C) \cdot 2}}{2 e^{- 2 x} \cdot 2}$

Step 13.2.18

Simplify the expression.

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Step 13.2.18.1

Multiply $2$ by $2$ .

$y = \pm \frac{\sqrt{(- 4 x e^{- 4 x} - e^{- 4 x} + 8 C) \cdot 2}}{4 e^{- 2 x}}$

Step 13.2.18.2

Reorder factors in $\pm \frac{\sqrt{(- 4 x e^{- 4 x} - e^{- 4 x} + 8 C) \cdot 2}}{4 e^{- 2 x}}$ .

$y = \pm \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

Step 13.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 13.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

Step 13.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

Step 13.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

$y = - \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

$y = \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

$y = - \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

$y = \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

$y = - \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + 8 C)}}{4 e^{- 2 x}}$

Step 14

Simplify the constant of integration.

$y = \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + C)}}{4 e^{- 2 x}}$

$y = - \frac{\sqrt{2 (- 4 x e^{- 4 x} - e^{- 4 x} + C)}}{4 e^{- 2 x}}$