Calculus Examples

$\frac{d y}{d x} = \frac{x^{2}}{10 y} - \frac{2 x}{5 y}$

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

To write $- \frac{2 x}{5 y}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d y}{d x} = \frac{x^{2}}{10 y} - \frac{2 x}{5 y} \cdot \frac{2}{2}$

Step 1.1.2

Write each expression with a common denominator of $10 y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.1

Multiply $\frac{2 x}{5 y}$ by $\frac{2}{2}$ .

$\frac{d y}{d x} = \frac{x^{2}}{10 y} - \frac{2 x \cdot 2}{5 y \cdot 2}$

Step 1.1.2.2

Multiply $2$ by $5$ .

$\frac{d y}{d x} = \frac{x^{2}}{10 y} - \frac{2 x \cdot 2}{10 y}$

Step 1.1.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{x^{2} - 2 x \cdot 2}{10 y}$

Step 1.1.4

Simplify the numerator.

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Step 1.1.4.1

Factor $x$ out of $x^{2} - 2 x \cdot 2$ .

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Step 1.1.4.1.1

Factor $x$ out of $x^{2}$ .

$\frac{d y}{d x} = \frac{x \cdot x - 2 x \cdot 2}{10 y}$

Step 1.1.4.1.2

Factor $x$ out of $- 2 x \cdot 2$ .

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

Step 1.1.4.1.3

Factor $x$ out of $x \cdot x + x (- 2 \cdot 2)$ .

$\frac{d y}{d x} = \frac{x (x - 2 \cdot 2)}{10 y}$

Step 1.1.4.2

Multiply $- 2$ by $2$ .

$\frac{d y}{d x} = \frac{x (x - 4)}{10 y}$

Step 1.2

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{x (x - 4)}{10 y}$

Step 1.3

Cancel the common factor of $y$ .

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Step 1.3.1

Factor $y$ out of $10 y$ .

$y \frac{d y}{d x} = y \frac{x (x - 4)}{y \cdot 10}$

Step 1.3.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x (x - 4)}{y \cdot 10}$

Step 1.3.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x (x - 4)}{10}$

Step 1.4

Rewrite the equation.

$y d y = \frac{x (x - 4)}{10} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x (x - 4)}{10} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x (x - 4)}{10} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{10}$ is constant with respect to $x$ , move $\frac{1}{10}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x (x - 4) d x$

Step 2.3.2

Multiply $x (x - 4)$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x \cdot x + x \cdot - 4 d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x^{1} x + x \cdot - 4 d x$

Step 2.3.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x^{1} x^{1} + x \cdot - 4 d x$

Step 2.3.3.3

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x^{1 + 1} + x \cdot - 4 d x$

Step 2.3.3.4

Add $1$ and $1$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x^{2} + x \cdot - 4 d x$

Step 2.3.3.5

Move $- 4$ to the left of $x$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} \int x^{2} - 4 x d x$

Step 2.3.4

Split the single integral into multiple integrals.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\int x^{2} d x + \int - 4 x d x)$

Step 2.3.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\frac{1}{3} x^{3} + C_{2} + \int - 4 x d x)$

Step 2.3.6

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\frac{1}{3} x^{3} + C_{2} - 4 \int x d x)$

Step 2.3.7

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\frac{1}{3} x^{3} + C_{2} - 4 (\frac{1}{2} x^{2} + C_{3}))$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\frac{x^{3}}{3} - 2 x^{2}) + C_{4}$

Step 2.3.8.2

Reorder terms.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{10} (\frac{1}{3} x^{3} - 2 x^{2}) + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{10} (\frac{1}{3} x^{3} - 2 x^{2}) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 3.2.2.1.1.2

Apply the distributive property.

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

Step 3.2.2.1.1.3

Combine.

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

Step 3.2.2.1.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.4.1

Factor $2$ out of $10$ .

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

Step 3.2.2.1.1.4.2

Factor $2$ out of $- 2 x^{2}$ .

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

Step 3.2.2.1.1.4.3

Cancel the common factor.

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

Step 3.2.2.1.1.4.4

Rewrite the expression.

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

Step 3.2.2.1.1.5

Combine $\frac{1}{5}$ and $x^{2}$ .

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

Step 3.2.2.1.1.6

Simplify each term.

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Step 3.2.2.1.1.6.1

Multiply $x^{3}$ by $1$ .

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

Step 3.2.2.1.1.6.2

Multiply $10$ by $3$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Factor $2$ out of $30$ .

$y^{2} = 2 \frac{x^{3}}{2 (15)} + 2 (- \frac{x^{2}}{5}) + 2 K$

Step 3.2.2.1.3.1.2

Cancel the common factor.

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

Step 3.2.2.1.3.1.3

Rewrite the expression.

$y^{2} = \frac{x^{3}}{15} + 2 (- \frac{x^{2}}{5}) + 2 K$

Step 3.2.2.1.3.2

Multiply $2 (- \frac{x^{2}}{5})$ .

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Step 3.2.2.1.3.2.1

Multiply $- 1$ by $2$ .

$y^{2} = \frac{x^{3}}{15} - 2 \frac{x^{2}}{5} + 2 K$

Step 3.2.2.1.3.2.2

Combine $- 2$ and $\frac{x^{2}}{5}$ .

$y^{2} = \frac{x^{3}}{15} + \frac{- 2 x^{2}}{5} + 2 K$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$y^{2} = \frac{x^{3}}{15} - \frac{2 x^{2}}{5} + 2 K$

Step 3.3

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

$y = \pm \sqrt{\frac{x^{3}}{15} - \frac{2 x^{2}}{5} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{\frac{x^{3}}{15} - \frac{2 x^{2}}{5} + 2 K}$ .

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Step 3.4.1

To write $- \frac{2 x^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x^{3}}{15} - \frac{2 x^{2}}{5} \cdot \frac{3}{3} + 2 K}$

Step 3.4.2

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

Multiply $\frac{2 x^{2}}{5}$ by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x^{3}}{15} - \frac{2 x^{2} \cdot 3}{5 \cdot 3} + 2 K}$

Step 3.4.2.2

Multiply $5$ by $3$ .

$y = \pm \sqrt{\frac{x^{3}}{15} - \frac{2 x^{2} \cdot 3}{15} + 2 K}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{3} - 2 x^{2} \cdot 3}{15} + 2 K}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Factor $x^{2}$ out of $x^{3} - 2 x^{2} \cdot 3$ .

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Step 3.4.4.1.1

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

$y = \pm \sqrt{\frac{x^{2} x - 2 x^{2} \cdot 3}{15} + 2 K}$

Step 3.4.4.1.2

Factor $x^{2}$ out of $- 2 x^{2} \cdot 3$ .

$y = \pm \sqrt{\frac{x^{2} x + x^{2} (- 2 \cdot 3)}{15} + 2 K}$

Step 3.4.4.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 2 \cdot 3)$ .

$y = \pm \sqrt{\frac{x^{2} (x - 2 \cdot 3)}{15} + 2 K}$

Step 3.4.4.2

Multiply $- 2$ by $3$ .

$y = \pm \sqrt{\frac{x^{2} (x - 6)}{15} + 2 K}$

Step 3.4.5

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$y = \pm \sqrt{\frac{x^{2} (x - 6)}{15} + 2 K \cdot \frac{15}{15}}$

Step 3.4.6

Simplify terms.

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Step 3.4.6.1

Combine $2 K$ and $\frac{15}{15}$ .

$y = \pm \sqrt{\frac{x^{2} (x - 6)}{15} + \frac{2 K \cdot 15}{15}}$

Step 3.4.6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} (x - 6) + 2 K \cdot 15}{15}}$

Step 3.4.7

Simplify the numerator.

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Step 3.4.7.1

Apply the distributive property.

$y = \pm \sqrt{\frac{x^{2} x + x^{2} \cdot - 6 + 2 K \cdot 15}{15}}$

Step 3.4.7.2

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

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Step 3.4.7.2.1

Multiply $x^{2}$ by $x$ .

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Step 3.4.7.2.1.1

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

$y = \pm \sqrt{\frac{x^{2} x^{1} + x^{2} \cdot - 6 + 2 K \cdot 15}{15}}$

Step 3.4.7.2.1.2

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

$y = \pm \sqrt{\frac{x^{2 + 1} + x^{2} \cdot - 6 + 2 K \cdot 15}{15}}$

Step 3.4.7.2.2

Add $2$ and $1$ .

$y = \pm \sqrt{\frac{x^{3} + x^{2} \cdot - 6 + 2 K \cdot 15}{15}}$

Step 3.4.7.3

Move $- 6$ to the left of $x^{2}$ .

$y = \pm \sqrt{\frac{x^{3} - 6 \cdot x^{2} + 2 K \cdot 15}{15}}$

Step 3.4.7.4

Multiply $15$ by $2$ .

$y = \pm \sqrt{\frac{x^{3} - 6 x^{2} + 30 K}{15}}$

Step 3.4.8

Rewrite $\sqrt{\frac{x^{3} - 6 x^{2} + 30 K}{15}}$ as $\frac{\sqrt{x^{3} - 6 x^{2} + 30 K}}{\sqrt{15}}$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K}}{\sqrt{15}}$

Step 3.4.9

Multiply $\frac{\sqrt{x^{3} - 6 x^{2} + 30 K}}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K}}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Step 3.4.10

Combine and simplify the denominator.

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Step 3.4.10.1

Multiply $\frac{\sqrt{x^{3} - 6 x^{2} + 30 K}}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 3.4.10.2

Raise $\sqrt{15}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{{\sqrt{15}}^{1} \sqrt{15}}$

Step 3.4.10.3

Raise $\sqrt{15}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{{\sqrt{15}}^{1} {\sqrt{15}}^{1}}$

Step 3.4.10.4

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

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 3.4.10.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{{\sqrt{15}}^{2}}$

Step 3.4.10.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 3.4.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 3.4.10.6.2

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

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 3.4.10.6.3

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

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{15^{\frac{2}{2}}}$

Step 3.4.10.6.4

Cancel the common factor of $2$ .

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Step 3.4.10.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{15^{\frac{2}{2}}}$

Step 3.4.10.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{15^{1}}$

Step 3.4.10.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x^{3} - 6 x^{2} + 30 K} \sqrt{15}}{15}$

Step 3.4.11

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x^{3} - 6 x^{2} + 30 K) \cdot 15}}{15}$

Step 3.4.12

Reorder factors in $\pm \frac{\sqrt{(x^{3} - 6 x^{2} + 30 K) \cdot 15}}{15}$ .

$y = \pm \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

Step 3.5

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

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Step 3.5.1

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

$y = \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

Step 3.5.2

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

$y = - \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

Step 3.5.3

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

$y = \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

$y = - \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

$y = \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

$y = - \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

$y = \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

$y = - \frac{\sqrt{15 (x^{3} - 6 x^{2} + 30 K)}}{15}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{15 (x^{3} - 6 x^{2} + K)}}{15}$

$y = - \frac{\sqrt{15 (x^{3} - 6 x^{2} + K)}}{15}$