Calculus Examples

$(x + 4 y^{2}) d y + 2 (y d) x = 0$

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

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Step 1.1

Rewrite.

$2 y d x + (x + 4 y^{2}) d y = 0$

Step 2

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 2 y$ .

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Step 2.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 y]$

Step 2.2

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

$\frac{\partial M}{\partial y} = 2 \frac{d}{d y} [y]$

Step 2.3

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

$\frac{\partial M}{\partial y} = 2 \cdot 1$

Step 2.4

Multiply $2$ by $1$ .

$\frac{\partial M}{\partial y} = 2$

Step 3

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x + 4 y^{2}$ .

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Step 3.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x + 4 y^{2}]$

Step 3.2

By the Sum Rule, the derivative of $x + 4 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [4 y^{2}]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x] + \frac{d}{d x} [4 y^{2}]$

Step 3.3

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

$\frac{\partial N}{\partial x} = 1 + \frac{d}{d x} [4 y^{2}]$

Step 3.4

Since $4 y^{2}$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 1 + 0$

Step 3.5

Add $1$ and $0$ .

$\frac{\partial N}{\partial x} = 1$

Step 4

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

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Step 4.1

Substitute $2$ for $\frac{\partial M}{\partial y}$ and $1$ for $\frac{\partial N}{\partial x}$ .

$2 = 1$

Step 4.2

Since the left side does not equal the right side, the equation is not an identity.

$2 = 1$ is not an identity.

Step 5

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

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Step 5.1

Substitute $1$ for $\frac{\partial N}{\partial x}$ .

$\frac{1 - \frac{\partial M}{\partial y}}{M}$

Step 5.2

Substitute $2$ for $\frac{\partial M}{\partial y}$ .

$\frac{1 - 2}{M}$

Step 5.3

Substitute $2 y$ for $M$ .

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Step 5.3.1

Substitute $2 y$ for $M$ .

$\frac{1 - 2}{2 y}$

Step 5.3.2

Subtract $2$ from $1$ .

$\frac{- 1}{2 y}$

Step 5.3.3

Substitute $2 y$ for $M$ .

$- \frac{1}{2 y}$

Step 5.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} d y}$ .

$μ (x, y) = e^{\int - \frac{1}{2 y} d y}$

Step 6

Evaluate the integral $e^{\int - \frac{1}{2 y} d y}$ .

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Step 6.1

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

$μ (x, y) = e^{- \int \frac{1}{2 y} d y}$

Step 6.2

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

$μ (x, y) = e^{- (\frac{1}{2} \int \frac{1}{y} d y)}$

Step 6.3

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

$μ (x, y) = e^{- \frac{1}{2} (\ln (| y |) + C)}$

Step 6.4

Simplify.

$μ (x, y) = e^{- \frac{1}{2} \ln (| y |)} + C$

Step 6.5

Simplify each term.

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Step 6.5.1

Multiply $- \frac{1}{2} \ln (| y |)$ .

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Step 6.5.1.1

Reorder $\ln (| y |)$ and $\frac{1}{2}$ .

$μ (x, y) = e^{- (\frac{1}{2} \ln (| y |))} + C$

Step 6.5.1.2

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

$μ (x, y) = e^{- \ln ({| y |}^{\frac{1}{2}})} + C$

Step 6.5.2

Simplify $- \ln ({| y |}^{\frac{1}{2}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| y |}^{\frac{1}{2}})}^{- 1})} + C$

Step 6.5.3

Exponentiation and log are inverse functions.

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

Step 6.5.4

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

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Step 6.5.4.1

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

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

Step 6.5.4.2

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

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

Step 6.5.4.3

Move the negative in front of the fraction.

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

Step 6.5.5

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

$μ (x, y) = \frac{1}{{| y |}^{\frac{1}{2}}} + C$

Step 7

Multiply both sides of $2 y d x + (x + 4 y^{2}) d y = 0$ by the integration factor $\frac{1}{y^{\frac{1}{2}}}$ .

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Step 7.1

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

$2 y \frac{1}{y^{\frac{1}{2}}} d x + (x + 4 y^{2}) d y = 0$

Step 7.2

Multiply $2 y \frac{1}{y^{\frac{1}{2}}}$ .

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Step 7.2.1

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

$y \frac{2}{y^{\frac{1}{2}}} d x + (x + 4 y^{2}) d y = 0$

Step 7.2.2

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

$\frac{y \cdot 2}{y^{\frac{1}{2}}} d x + (x + 4 y^{2}) d y = 0$

Step 7.3

Move $y^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$y \cdot 2 y^{- \frac{1}{2}} d x + (x + 4 y^{2}) d y = 0$

Step 7.4

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

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Step 7.4.1

Move $y^{- \frac{1}{2}}$ .

$y^{- \frac{1}{2}} y \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.4.2

Multiply $y^{- \frac{1}{2}}$ by $y$ .

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Step 7.4.2.1

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

$y^{- \frac{1}{2}} y^{1} \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.4.2.2

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

$y^{- \frac{1}{2} + 1} \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.4.3

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

$y^{- \frac{1}{2} + \frac{2}{2}} \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.4.4

Combine the numerators over the common denominator.

$y^{\frac{- 1 + 2}{2}} \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.4.5

Add $- 1$ and $2$ .

$y^{\frac{1}{2}} \cdot 2 d x + (x + 4 y^{2}) d y = 0$

Step 7.5

Move $2$ to the left of $y^{\frac{1}{2}}$ .

$2 y^{\frac{1}{2}} d x + (x + 4 y^{2}) d y = 0$

Step 7.6

Multiply $x + 4 y^{2}$ by $\frac{1}{y^{\frac{1}{2}}}$ .

$2 y^{\frac{1}{2}} d x + (x + 4 y^{2}) \frac{1}{y^{\frac{1}{2}}} d y = 0$

Step 7.7

Multiply $x + 4 y^{2}$ by $\frac{1}{y^{\frac{1}{2}}}$ .

$2 y^{\frac{1}{2}} d x + \frac{x + 4 y^{2}}{y^{\frac{1}{2}}} d y = 0$

Step 8

Set $f (x, y)$ equal to the integral of $M (x, y)$ .

$f (x, y) = \int 2 y^{\frac{1}{2}} d x$

Step 9

Integrate $M (x, y) = 2 y^{\frac{1}{2}}$ to find $f (x, y)$ .

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Step 9.1

Apply the constant rule.

$f (x, y) = 2 y^{\frac{1}{2}} x + C$

Step 10

Since the integral of $g (y)$ will contain an integration constant, we can replace $C$ with $g (y)$ .

$f (x, y) = 2 y^{\frac{1}{2}} x + g (y)$

Step 11

Set $\frac{\partial f}{\partial y} = N (x, y)$ .

$\frac{\partial f}{\partial y} = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12

Find $\frac{\partial f}{\partial y}$ .

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Step 12.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [2 y^{\frac{1}{2}} x + g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.2

By the Sum Rule, the derivative of $2 y^{\frac{1}{2}} x + g (y)$ with respect to $y$ is $\frac{d}{d y} [2 y^{\frac{1}{2}} x] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [2 y^{\frac{1}{2}} x] + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3

Evaluate $\frac{d}{d y} [2 y^{\frac{1}{2}} x]$ .

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Step 12.3.1

Since $2 x$ is constant with respect to $y$ , the derivative of $2 y^{\frac{1}{2}} x$ with respect to $y$ is $2 x \frac{d}{d y} [y^{\frac{1}{2}}]$ .

$2 x \frac{d}{d y} [y^{\frac{1}{2}}] + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.2

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

$2 x (\frac{1}{2} y^{\frac{1}{2} - 1}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x (\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.4

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

$2 x (\frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.5

Combine the numerators over the common denominator.

$2 x (\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.6

Simplify the numerator.

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Step 12.3.6.1

Multiply $- 1$ by $2$ .

$2 x (\frac{1}{2} y^{\frac{1 - 2}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.6.2

Subtract $2$ from $1$ .

$2 x (\frac{1}{2} y^{\frac{- 1}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.7

Move the negative in front of the fraction.

$2 x (\frac{1}{2} y^{- \frac{1}{2}}) + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.8

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

$2 x \frac{y^{- \frac{1}{2}}}{2} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.9

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

$x \frac{2 y^{- \frac{1}{2}}}{2} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.10

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

$\frac{x (2 y^{- \frac{1}{2}})}{2} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.11

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

$\frac{2 \cdot x y^{- \frac{1}{2}}}{2} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.12

Move $y^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 x}{2 y^{\frac{1}{2}}} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.13

Cancel the common factor.

$\frac{2 x}{2 y^{\frac{1}{2}}} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.3.14

Rewrite the expression.

$\frac{x}{y^{\frac{1}{2}}} + \frac{d}{d y} [g (y)] = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.4

Differentiate using the function rule which states that the derivative of $g (y)$ is $\frac{d g}{d y}$ .

$\frac{x}{y^{\frac{1}{2}}} + \frac{d g}{d y} = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 12.5

Reorder terms.

$\frac{d g}{d y} + \frac{x}{y^{\frac{1}{2}}} = \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$

Step 13

Solve for $\frac{d g}{d y}$ .

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Step 13.1

Solve for $\frac{d g}{d y}$ .

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Step 13.1.1

Simplify $\frac{d g}{d y} + \frac{x}{y^{\frac{1}{2}}} - \frac{x + 4 y^{2}}{y^{\frac{1}{2}}}$ .

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Step 13.1.1.1

Combine the numerators over the common denominator.

$\frac{d g}{d y} + \frac{x - (x + 4 y^{2})}{y^{\frac{1}{2}}} = 0$

Step 13.1.1.2

Simplify each term.

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Step 13.1.1.2.1

Apply the distributive property.

$\frac{d g}{d y} + \frac{x - x - (4 y^{2})}{y^{\frac{1}{2}}} = 0$

Step 13.1.1.2.2

Multiply $4$ by $- 1$ .

$\frac{d g}{d y} + \frac{x - x - 4 y^{2}}{y^{\frac{1}{2}}} = 0$

Step 13.1.1.3

Simplify by adding terms.

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Step 13.1.1.3.1

Subtract $x$ from $x$ .

$\frac{d g}{d y} + \frac{0 - 4 y^{2}}{y^{\frac{1}{2}}} = 0$

Step 13.1.1.3.2

Subtract $4 y^{2}$ from $0$ .

$\frac{d g}{d y} + \frac{- 4 y^{2}}{y^{\frac{1}{2}}} = 0$

Step 13.1.1.4

Simplify each term.

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Step 13.1.1.4.1

Move $y^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{d g}{d y} - 4 y^{2} y^{- \frac{1}{2}} = 0$

Step 13.1.1.4.2

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

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Step 13.1.1.4.2.1

Move $y^{- \frac{1}{2}}$ .

$\frac{d g}{d y} - 4 (y^{- \frac{1}{2}} y^{2}) = 0$

Step 13.1.1.4.2.2

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

$\frac{d g}{d y} - 4 y^{- \frac{1}{2} + 2} = 0$

Step 13.1.1.4.2.3

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

$\frac{d g}{d y} - 4 y^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} = 0$

Step 13.1.1.4.2.4

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

$\frac{d g}{d y} - 4 y^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} = 0$

Step 13.1.1.4.2.5

Combine the numerators over the common denominator.

$\frac{d g}{d y} - 4 y^{\frac{- 1 + 2 \cdot 2}{2}} = 0$

Step 13.1.1.4.2.6

Simplify the numerator.

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Step 13.1.1.4.2.6.1

Multiply $2$ by $2$ .

$\frac{d g}{d y} - 4 y^{\frac{- 1 + 4}{2}} = 0$

Step 13.1.1.4.2.6.2

Add $- 1$ and $4$ .

$\frac{d g}{d y} - 4 y^{\frac{3}{2}} = 0$

Step 13.1.2

Add $4 y^{\frac{3}{2}}$ to both sides of the equation.

$\frac{d g}{d y} = 4 y^{\frac{3}{2}}$

Step 14

Find the antiderivative of $4 y^{\frac{3}{2}}$ to find $g (y)$ .

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Step 14.1

Integrate both sides of $\frac{d g}{d y} = 4 y^{\frac{3}{2}}$ .

$\int \frac{d g}{d y} d y = \int 4 y^{\frac{3}{2}} d y$

Step 14.2

Evaluate $\int \frac{d g}{d y} d y$ .

$g (y) = \int 4 y^{\frac{3}{2}} d y$

Step 14.3

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$g (y) = 4 \int y^{\frac{3}{2}} d y$

Step 14.4

By the Power Rule, the integral of $y^{\frac{3}{2}}$ with respect to $y$ is $\frac{2}{5} y^{\frac{5}{2}}$ .

$g (y) = 4 (\frac{2}{5} y^{\frac{5}{2}} + C)$

Step 14.5

Simplify the answer.

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Step 14.5.1

Rewrite $4 (\frac{2}{5} y^{\frac{5}{2}} + C)$ as $4 (\frac{2}{5}) y^{\frac{5}{2}} + C$ .

$g (y) = 4 (\frac{2}{5}) y^{\frac{5}{2}} + C$

Step 14.5.2

Simplify.

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Step 14.5.2.1

Combine $4$ and $\frac{2}{5}$ .

$g (y) = \frac{4 \cdot 2}{5} y^{\frac{5}{2}} + C$

Step 14.5.2.2

Multiply $4$ by $2$ .

$g (y) = \frac{8}{5} y^{\frac{5}{2}} + C$

Step 15

Substitute for $g (y)$ in $f (x, y) = 2 y^{\frac{1}{2}} x + g (y)$ .

$f (x, y) = 2 y^{\frac{1}{2}} x + \frac{8}{5} y^{\frac{5}{2}} + C$

Step 16

Simplify $f (x, y) = 2 y^{\frac{1}{2}} x + \frac{8}{5} y^{\frac{5}{2}} + C$ .

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Step 16.1

Combine $\frac{8}{5}$ and $y^{\frac{5}{2}}$ .

$f (x, y) = 2 y^{\frac{1}{2}} x + \frac{8 y^{\frac{5}{2}}}{5} + C$

Step 16.2

Reorder factors in $f (x, y) = 2 y^{\frac{1}{2}} x + \frac{8 y^{\frac{5}{2}}}{5} + C$ .

$f (x, y) = 2 x y^{\frac{1}{2}} + \frac{8 y^{\frac{5}{2}}}{5} + C$