Calculus Examples

$\frac{d y}{d x} = 7 x \sqrt{2 y + 20}$ , $y (0) = 5$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{2 y + 20}}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{1}{\sqrt{2 y + 20}} (7 x \sqrt{2 y + 20})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{1}{\sqrt{2 y + 20}} (x \sqrt{2 y + 20})$

Step 1.2.2

Factor $2$ out of $2 y + 20$ .

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Step 1.2.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{1}{\sqrt{2 (y) + 20}} (x \sqrt{2 y + 20})$

Step 1.2.2.2

Factor $2$ out of $20$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{1}{\sqrt{2 y + 2 \cdot 10}} (x \sqrt{2 y + 20})$

Step 1.2.2.3

Factor $2$ out of $2 y + 2 \cdot 10$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{1}{\sqrt{2 (y + 10)}} (x \sqrt{2 y + 20})$

Step 1.2.3

Multiply $\frac{1}{\sqrt{2 (y + 10)}}$ by $\frac{\sqrt{2 (y + 10)}}{\sqrt{2 (y + 10)}}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 (\frac{1}{\sqrt{2 (y + 10)}} \cdot \frac{\sqrt{2 (y + 10)}}{\sqrt{2 (y + 10)}}) (x \sqrt{2 y + 20})$

Step 1.2.4

Combine and simplify the denominator.

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Step 1.2.4.1

Multiply $\frac{1}{\sqrt{2 (y + 10)}}$ by $\frac{\sqrt{2 (y + 10)}}{\sqrt{2 (y + 10)}}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{\sqrt{2 (y + 10)} \sqrt{2 (y + 10)}} (x \sqrt{2 y + 20})$

Step 1.2.4.2

Raise $\sqrt{2 (y + 10)}$ to the power of $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{\sqrt{2 (y + 10)}}^{1} \sqrt{2 (y + 10)}} (x \sqrt{2 y + 20})$

Step 1.2.4.3

Raise $\sqrt{2 (y + 10)}$ to the power of $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{\sqrt{2 (y + 10)}}^{1} {\sqrt{2 (y + 10)}}^{1}} (x \sqrt{2 y + 20})$

Step 1.2.4.4

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{\sqrt{2 (y + 10)}}^{1 + 1}} (x \sqrt{2 y + 20})$

Step 1.2.4.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{\sqrt{2 (y + 10)}}^{2}} (x \sqrt{2 y + 20})$

Step 1.2.4.6

Rewrite ${\sqrt{2 (y + 10)}}^{2}$ as $2 (y + 10)$ .

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Step 1.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (y + 10)}$ as ${(2 (y + 10))}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{({(2 (y + 10))}^{\frac{1}{2}})}^{2}} (x \sqrt{2 y + 20})$

Step 1.2.4.6.2

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{(2 (y + 10))}^{\frac{1}{2} \cdot 2}} (x \sqrt{2 y + 20})$

Step 1.2.4.6.3

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{(2 (y + 10))}^{\frac{2}{2}}} (x \sqrt{2 y + 20})$

Step 1.2.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.4.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{(2 (y + 10))}^{\frac{2}{2}}} (x \sqrt{2 y + 20})$

Step 1.2.4.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{{(2 (y + 10))}^{1}} (x \sqrt{2 y + 20})$

Step 1.2.4.6.5

Simplify.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 \frac{\sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 y + 20})$

Step 1.2.5

Combine $7$ and $\frac{\sqrt{2 (y + 10)}}{2 (y + 10)}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 y + 20})$

Step 1.2.6

Factor $2$ out of $2 y + 20$ .

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Step 1.2.6.1

Factor $2$ out of $2 y$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 (y) + 20})$

Step 1.2.6.2

Factor $2$ out of $20$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 y + 2 \cdot 10})$

Step 1.2.6.3

Factor $2$ out of $2 y + 2 \cdot 10$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 (y + 10)})$

Step 1.2.7

Multiply $\frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)} (x \sqrt{2 (y + 10)})$ .

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Step 1.2.7.1

Combine $x$ and $\frac{7 \sqrt{2 (y + 10)}}{2 (y + 10)}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 \sqrt{2 (y + 10)})}{2 (y + 10)} \sqrt{2 (y + 10)}$

Step 1.2.7.2

Combine $\frac{x (7 \sqrt{2 (y + 10)})}{2 (y + 10)}$ and $\sqrt{2 (y + 10)}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 \sqrt{2 (y + 10)}) \sqrt{2 (y + 10)}}{2 (y + 10)}$

Step 1.2.7.3

Raise $\sqrt{2 (y + 10)}$ to the power of $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 ({\sqrt{2 (y + 10)}}^{1} \sqrt{2 (y + 10)}))}{2 (y + 10)}$

Step 1.2.7.4

Raise $\sqrt{2 (y + 10)}$ to the power of $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 ({\sqrt{2 (y + 10)}}^{1} {\sqrt{2 (y + 10)}}^{1}))}{2 (y + 10)}$

Step 1.2.7.5

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 {\sqrt{2 (y + 10)}}^{1 + 1})}{2 (y + 10)}$

Step 1.2.7.6

Add $1$ and $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x (7 {\sqrt{2 (y + 10)}}^{2})}{2 (y + 10)}$

Step 1.2.8

Simplify the numerator.

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Step 1.2.8.1

Rewrite ${\sqrt{2 (y + 10)}}^{2}$ as $2 (y + 10)$ .

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Step 1.2.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (y + 10)}$ as ${(2 (y + 10))}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 {({(2 (y + 10))}^{\frac{1}{2}})}^{2}}{2 (y + 10)}$

Step 1.2.8.1.2

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 {(2 (y + 10))}^{\frac{1}{2} \cdot 2}}{2 (y + 10)}$

Step 1.2.8.1.3

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

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 {(2 (y + 10))}^{\frac{2}{2}}}{2 (y + 10)}$

Step 1.2.8.1.4

Cancel the common factor of $2$ .

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Step 1.2.8.1.4.1

Cancel the common factor.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 {(2 (y + 10))}^{\frac{2}{2}}}{2 (y + 10)}$

Step 1.2.8.1.4.2

Rewrite the expression.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 {(2 (y + 10))}^{1}}{2 (y + 10)}$

Step 1.2.8.1.5

Simplify.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 (y + 10))}{2 (y + 10)}$

Step 1.2.8.2

Apply the distributive property.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 y + 2 \cdot 10)}{2 (y + 10)}$

Step 1.2.8.3

Multiply $2$ by $10$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 y + 20)}{2 (y + 10)}$

Step 1.2.8.4

Factor $2$ out of $2 y + 20$ .

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Step 1.2.8.4.1

Factor $2$ out of $2 y$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 (y) + 20)}{2 (y + 10)}$

Step 1.2.8.4.2

Factor $2$ out of $20$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 y + 2 \cdot 10)}{2 (y + 10)}$

Step 1.2.8.4.3

Factor $2$ out of $2 y + 2 \cdot 10$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (2 (y + 10))}{2 (y + 10)}$

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 \cdot 2 (y + 10)}{2 (y + 10)}$

Step 1.2.8.5

Multiply $2$ by $7$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 14 (y + 10)}{2 (y + 10)}$

Step 1.2.9

Cancel the common factor of $14$ and $2$ .

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Step 1.2.9.1

Factor $2$ out of $x \cdot 14 (y + 10)$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{2 (x \cdot 7 (y + 10))}{2 (y + 10)}$

Step 1.2.9.2

Cancel the common factors.

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Step 1.2.9.2.1

Cancel the common factor.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{2 (x \cdot 7 (y + 10))}{2 (y + 10)}$

Step 1.2.9.2.2

Rewrite the expression.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (y + 10)}{y + 10}$

Step 1.2.10

Cancel the common factor of $y + 10$ .

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Step 1.2.10.1

Cancel the common factor.

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = \frac{x \cdot 7 (y + 10)}{y + 10}$

Step 1.2.10.2

Divide $x \cdot 7$ by $1$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = x \cdot 7$

Step 1.2.11

Move $7$ to the left of $x$ .

$\frac{1}{\sqrt{2 y + 20}} \frac{d y}{d x} = 7 x$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{2 y + 20}} d y = 7 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{2 y + 20}} d y = \int 7 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 2 y + 20$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 2 y + 20$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $2 y + 20$ .

$\frac{d}{d y} [2 y + 20]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $2 y + 20$ with respect to $y$ is $\frac{d}{d y} [2 y] + \frac{d}{d y} [20]$ .

$\frac{d}{d y} [2 y] + \frac{d}{d y} [20]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [2 y]$ .

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Step 2.2.1.1.3.1

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

$2 \frac{d}{d y} [y] + \frac{d}{d y} [20]$

Step 2.2.1.1.3.2

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

$2 \cdot 1 + \frac{d}{d y} [20]$

Step 2.2.1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d y} [20]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

Since $20$ is constant with respect to $y$ , the derivative of $20$ with respect to $y$ is $0$ .

$2 + 0$

Step 2.2.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.1.2

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

$\int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} d u = \int 7 x d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

$\int \frac{1}{\sqrt{u} \cdot 2} d u = \int 7 x d x$

Step 2.2.2.2

Move $2$ to the left of $\sqrt{u}$ .

$\int \frac{1}{2 \sqrt{u}} d u = \int 7 x d x$

Step 2.2.3

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

$\frac{1}{2} \int \frac{1}{\sqrt{u}} d u = \int 7 x d x$

Step 2.2.4

Apply basic rules of exponents.

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Step 2.2.4.1

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

$\frac{1}{2} \int \frac{1}{u^{\frac{1}{2}}} d u = \int 7 x d x$

Step 2.2.4.2

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

$\frac{1}{2} \int {(u^{\frac{1}{2}})}^{- 1} d u = \int 7 x d x$

Step 2.2.4.3

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

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Step 2.2.4.3.1

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

$\frac{1}{2} \int u^{\frac{1}{2} \cdot - 1} d u = \int 7 x d x$

Step 2.2.4.3.2

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

$\frac{1}{2} \int u^{\frac{- 1}{2}} d u = \int 7 x d x$

Step 2.2.4.3.3

Move the negative in front of the fraction.

$\frac{1}{2} \int u^{- \frac{1}{2}} d u = \int 7 x d x$

Step 2.2.5

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

$\frac{1}{2} (2 u^{\frac{1}{2}} + C_{1}) = \int 7 x d x$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $\frac{1}{2} (2 u^{\frac{1}{2}} + C_{1})$ as $\frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C_{1}$ .

$\frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.2.6.2

Simplify.

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Step 2.2.6.2.1

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

$\frac{2}{2} u^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.2.6.2.2

Cancel the common factor of $2$ .

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Step 2.2.6.2.2.1

Cancel the common factor.

$\frac{2}{2} u^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.2.6.2.2.2

Rewrite the expression.

$1 u^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.2.6.2.3

Multiply $u^{\frac{1}{2}}$ by $1$ .

$u^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.2.7

Replace all occurrences of $u$ with $2 y + 20$ .

${(2 y + 20)}^{\frac{1}{2}} + C_{1} = \int 7 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

${(2 y + 20)}^{\frac{1}{2}} + C_{1} = 7 \int x d x$

Step 2.3.2

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

${(2 y + 20)}^{\frac{1}{2}} + C_{1} = 7 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $7 (\frac{1}{2} x^{2} + C_{2})$ as $7 (\frac{1}{2}) x^{2} + C_{2}$ .

${(2 y + 20)}^{\frac{1}{2}} + C_{1} = 7 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Combine $7$ and $\frac{1}{2}$ .

${(2 y + 20)}^{\frac{1}{2}} + C_{1} = \frac{7}{2} x^{2} + C_{2}$

Step 2.4

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

${(2 y + 20)}^{\frac{1}{2}} = \frac{7}{2} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(2 y + 20)}^{\frac{1}{2}})}^{2} = {(\frac{7}{2} x^{2} + K)}^{2}$

Step 3.2

Simplify the exponent.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify ${({(2 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Multiply the exponents in ${({(2 y + 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1.1

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

${(2 y + 20)}^{\frac{1}{2} \cdot 2} = {(\frac{7}{2} x^{2} + K)}^{2}$

Step 3.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.2.1

Cancel the common factor.

${(2 y + 20)}^{\frac{1}{2} \cdot 2} = {(\frac{7}{2} x^{2} + K)}^{2}$

Step 3.2.1.1.1.2.2

Rewrite the expression.

${(2 y + 20)}^{1} = {(\frac{7}{2} x^{2} + K)}^{2}$

Step 3.2.1.1.2

Simplify.

$2 y + 20 = {(\frac{7}{2} x^{2} + K)}^{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify ${(\frac{7}{2} x^{2} + K)}^{2}$ .

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Step 3.2.2.1.1

Combine fractions.

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Step 3.2.2.1.1.1

Combine $\frac{7}{2}$ and $x^{2}$ .

$2 y + 20 = {(\frac{7 x^{2}}{2} + K)}^{2}$

Step 3.2.2.1.1.2

Rewrite ${(\frac{7 x^{2}}{2} + K)}^{2}$ as $(\frac{7 x^{2}}{2} + K) (\frac{7 x^{2}}{2} + K)$ .

$2 y + 20 = (\frac{7 x^{2}}{2} + K) (\frac{7 x^{2}}{2} + K)$

Step 3.2.2.1.2

Expand $(\frac{7 x^{2}}{2} + K) (\frac{7 x^{2}}{2} + K)$ using the FOIL Method.

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Step 3.2.2.1.2.1

Apply the distributive property.

$2 y + 20 = \frac{7 x^{2}}{2} (\frac{7 x^{2}}{2} + K) + K (\frac{7 x^{2}}{2} + K)$

Step 3.2.2.1.2.2

Apply the distributive property.

$2 y + 20 = \frac{7 x^{2}}{2} \cdot \frac{7 x^{2}}{2} + \frac{7 x^{2}}{2} K + K (\frac{7 x^{2}}{2} + K)$

Step 3.2.2.1.2.3

Apply the distributive property.

$2 y + 20 = \frac{7 x^{2}}{2} \cdot \frac{7 x^{2}}{2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3

Simplify and combine like terms.

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Step 3.2.2.1.3.1

Simplify each term.

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Step 3.2.2.1.3.1.1

Combine.

$2 y + 20 = \frac{7 x^{2} (7 x^{2})}{2 \cdot 2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2

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

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Step 3.2.2.1.3.1.2.1

Move $x^{2}$ .

$2 y + 20 = \frac{7 (x^{2} x^{2}) \cdot 7}{2 \cdot 2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2.2

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

$2 y + 20 = \frac{7 x^{2 + 2} \cdot 7}{2 \cdot 2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2.3

Add $2$ and $2$ .

$2 y + 20 = \frac{7 x^{4} \cdot 7}{2 \cdot 2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.3

Multiply $7$ by $7$ .

$2 y + 20 = \frac{49 x^{4}}{2 \cdot 2} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.4

Multiply $2$ by $2$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 x^{2}}{2} K + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.5

Combine $\frac{7 x^{2}}{2}$ and $K$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 x^{2} K}{2} + K \frac{7 x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.6

Combine $K$ and $\frac{7 x^{2}}{2}$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 x^{2} K}{2} + \frac{K (7 x^{2})}{2} + K (K)$

Step 3.2.2.1.3.1.7

Move $7$ to the left of $K$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 x^{2} K}{2} + \frac{7 \cdot K x^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.8

Multiply $K$ by $K$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 x^{2} K}{2} + \frac{7 K x^{2}}{2} + K^{2}$

Step 3.2.2.1.3.2

Add $\frac{7 x^{2} K}{2}$ and $\frac{7 K x^{2}}{2}$ .

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Step 3.2.2.1.3.2.1

Move $x^{2}$ .

$2 y + 20 = \frac{49 x^{4}}{4} + \frac{7 K x^{2}}{2} + \frac{7 K x^{2}}{2} + K^{2}$

Step 3.2.2.1.3.2.2

Add $\frac{7 K x^{2}}{2}$ and $\frac{7 K x^{2}}{2}$ .

$2 y + 20 = \frac{49 x^{4}}{4} + 2 \frac{7 K x^{2}}{2} + K^{2}$

Step 3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.4.1

Cancel the common factor.

$2 y + 20 = \frac{49 x^{4}}{4} + 2 \frac{7 K x^{2}}{2} + K^{2}$

Step 3.2.2.1.4.2

Rewrite the expression.

$2 y + 20 = \frac{49 x^{4}}{4} + 7 K x^{2} + K^{2}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Subtract $20$ from both sides of the equation.

$2 y = \frac{49 x^{4}}{4} + 7 K x^{2} + K^{2} - 20$

Step 3.3.2

Divide each term in $2 y = \frac{49 x^{4}}{4} + 7 K x^{2} + K^{2} - 20$ by $2$ and simplify.

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Step 3.3.2.1

Divide each term in $2 y = \frac{49 x^{4}}{4} + 7 K x^{2} + K^{2} - 20$ by $2$ .

$\frac{2 y}{2} = \frac{\frac{49 x^{4}}{4}}{2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\frac{49 x^{4}}{4}}{2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{49 x^{4}}{4}}{2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Simplify each term.

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Step 3.3.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{49 x^{4}}{4} \cdot \frac{1}{2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.3.1.2

Combine.

$y = \frac{49 x^{4} \cdot 1}{4 \cdot 2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.3.1.3

Multiply $49$ by $1$ .

$y = \frac{49 x^{4}}{4 \cdot 2} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.3.1.4

Multiply $4$ by $2$ .

$y = \frac{49 x^{4}}{8} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} + \frac{- 20}{2}$

Step 3.3.2.3.1.5

Divide $- 20$ by $2$ .

$y = \frac{49 x^{4}}{8} + \frac{7 K x^{2}}{2} + \frac{K^{2}}{2} - 10$

Step 4

Simplify the constant of integration.

$y = \frac{49 x^{4}}{8} + \frac{K x^{2}}{2} + K$

Step 5

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $5$ for $y$ in $y = \frac{49 x^{4}}{8} + \frac{K x^{2}}{2} + K$ .

$5 = \frac{49 \cdot 0^{4}}{8} + \frac{K \cdot 0^{2}}{2} + K$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\frac{49 \cdot 0^{4}}{8} + \frac{K \cdot 0^{2}}{2} + K = 5$ .

$\frac{49 \cdot 0^{4}}{8} + \frac{K \cdot 0^{2}}{2} + K = 5$

Step 6.2

Simplify $\frac{49 \cdot 0^{4}}{8} + \frac{K \cdot 0^{2}}{2} + K$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{49 \cdot 0}{8} + \frac{K \cdot 0^{2}}{2} + K = 5$

Step 6.2.1.2

Multiply $49$ by $0$ .

$\frac{0}{8} + \frac{K \cdot 0^{2}}{2} + K = 5$

Step 6.2.1.3

Divide $0$ by $8$ .

$0 + \frac{K \cdot 0^{2}}{2} + K = 5$

Step 6.2.1.4

Raising $0$ to any positive power yields $0$ .

$0 + \frac{K \cdot 0}{2} + K = 5$

Step 6.2.1.5

Multiply $K$ by $0$ .

$0 + \frac{0}{2} + K = 5$

Step 6.2.1.6

Divide $0$ by $2$ .

$0 + 0 + K = 5$

Step 6.2.2

Combine the opposite terms in $0 + 0 + K$ .

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Step 6.2.2.1

Add $0$ and $0$ .

$0 + K = 5$

Step 6.2.2.2

Add $0$ and $K$ .

$K = 5$

Step 7

Substitute $5$ for $K$ in $y = \frac{49 x^{4}}{8} + \frac{K x^{2}}{2} + K$ and simplify.

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Step 7.1

Substitute $5$ for $K$ .

$y = \frac{49 x^{4}}{8} + \frac{5 x^{2}}{2} + K$