Calculus Examples

$\frac{d y}{d x} + \frac{x^{2} + 25}{y^{3} - y^{2}} = 0$

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Simplify $\frac{d y}{d x} + \frac{x^{2} + 25}{y^{3} - y^{2}}$ .

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Step 1.1.1.1

Factor $y^{2}$ out of $y^{3} - y^{2}$ .

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Step 1.1.1.1.1

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

$\frac{d y}{d x} + \frac{x^{2} + 25}{y^{2} y - y^{2}} = 0$

Step 1.1.1.1.2

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

$\frac{d y}{d x} + \frac{x^{2} + 25}{y^{2} y + y^{2} \cdot - 1} = 0$

Step 1.1.1.1.3

Factor $y^{2}$ out of $y^{2} y + y^{2} \cdot - 1$ .

$\frac{d y}{d x} + \frac{x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.2

To write $\frac{d y}{d x}$ as a fraction with a common denominator, multiply by $\frac{(y - 1) y^{2}}{(y - 1) y^{2}}$ .

$\frac{d y}{d x} \cdot \frac{(y - 1) y^{2}}{(y - 1) y^{2}} + \frac{x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.3

Write each expression with a common denominator of $(y - 1) y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.1.3.1

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

$\frac{\frac{d y}{d x} ((y - 1) y^{2})}{(y - 1) y^{2}} + \frac{x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.3.2

Reorder the factors of $(y - 1) y^{2}$ .

$\frac{\frac{d y}{d x} ((y - 1) y^{2})}{y^{2} (y - 1)} + \frac{x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.4

Combine the numerators over the common denominator.

$\frac{\frac{d y}{d x} ((y - 1) y^{2}) + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5

Simplify the numerator.

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Step 1.1.1.5.1

Apply the distributive property.

$\frac{(\frac{d y}{d x} y + \frac{d y}{d x} \cdot - 1) y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.2

Move $- 1$ to the left of $\frac{d y}{d x}$ .

$\frac{(\frac{d y}{d x} y - 1 \cdot \frac{d y}{d x}) y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.3

Rewrite $- 1 \frac{d y}{d x}$ as $- \frac{d y}{d x}$ .

$\frac{(\frac{d y}{d x} y - \frac{d y}{d x}) y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.4

Apply the distributive property.

$\frac{\frac{d y}{d x} y \cdot y^{2} - \frac{d y}{d x} y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.5

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

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Step 1.1.1.5.5.1

Move $y^{2}$ .

$\frac{\frac{d y}{d x} (y^{2} y) - \frac{d y}{d x} y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.5.2

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

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Step 1.1.1.5.5.2.1

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

$\frac{\frac{d y}{d x} (y^{2} y^{1}) - \frac{d y}{d x} y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.5.2.2

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

$\frac{\frac{d y}{d x} y^{2 + 1} - \frac{d y}{d x} y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.1.5.5.3

Add $2$ and $1$ .

$\frac{\frac{d y}{d x} y^{3} - \frac{d y}{d x} y^{2} + x^{2} + 25}{y^{2} (y - 1)} = 0$

Step 1.1.2

Set the numerator equal to zero.

$\frac{d y}{d x} y^{3} - \frac{d y}{d x} y^{2} + x^{2} + 25 = 0$

Step 1.1.3

Solve the equation for $\frac{d y}{d x}$ .

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Step 1.1.3.1

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.3.1.1

Subtract $x^{2}$ from both sides of the equation.

$\frac{d y}{d x} y^{3} - \frac{d y}{d x} y^{2} + 25 = - x^{2}$

Step 1.1.3.1.2

Subtract $25$ from both sides of the equation.

$\frac{d y}{d x} y^{3} - \frac{d y}{d x} y^{2} = - x^{2} - 25$

Step 1.1.3.2

Factor $\frac{d y}{d x} y^{2}$ out of $\frac{d y}{d x} y^{3} - \frac{d y}{d x} y^{2}$ .

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Step 1.1.3.2.1

Factor $\frac{d y}{d x} y^{2}$ out of $\frac{d y}{d x} y^{3}$ .

$\frac{d y}{d x} y^{2} y - \frac{d y}{d x} y^{2} = - x^{2} - 25$

Step 1.1.3.2.2

Factor $\frac{d y}{d x} y^{2}$ out of $- \frac{d y}{d x} y^{2}$ .

$\frac{d y}{d x} y^{2} y + \frac{d y}{d x} y^{2} \cdot - 1 = - x^{2} - 25$

Step 1.1.3.2.3

Factor $\frac{d y}{d x} y^{2}$ out of $\frac{d y}{d x} y^{2} y + \frac{d y}{d x} y^{2} \cdot - 1$ .

$\frac{d y}{d x} y^{2} (y - 1) = - x^{2} - 25$

Step 1.1.3.3

Divide each term in $\frac{d y}{d x} y^{2} (y - 1) = - x^{2} - 25$ by $y^{2} (y - 1)$ and simplify.

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Step 1.1.3.3.1

Divide each term in $\frac{d y}{d x} y^{2} (y - 1) = - x^{2} - 25$ by $y^{2} (y - 1)$ .

$\frac{\frac{d y}{d x} y^{2} (y - 1)}{y^{2} (y - 1)} = \frac{- x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.2

Simplify the left side.

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Step 1.1.3.3.2.1

Cancel the common factor of $y^{2}$ .

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Step 1.1.3.3.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} y^{2} (y - 1)}{y^{2} (y - 1)} = \frac{- x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.2.1.2

Rewrite the expression.

$\frac{\frac{d y}{d x} (y - 1)}{y - 1} = \frac{- x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.2.2

Cancel the common factor of $y - 1$ .

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Step 1.1.3.3.2.2.1

Cancel the common factor.

$\frac{\frac{d y}{d x} (y - 1)}{y - 1} = \frac{- x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{- x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.1

Simplify each term.

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Step 1.1.3.3.3.1.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{x^{2}}{y^{2} (y - 1)} + \frac{- 25}{y^{2} (y - 1)}$

Step 1.1.3.3.3.1.2

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{x^{2}}{y^{2} (y - 1)} - \frac{25}{y^{2} (y - 1)}$

Step 1.2

Factor.

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Step 1.2.1

Factor $- 1$ out of $- \frac{x^{2}}{y^{2} (y - 1)} - \frac{25}{y^{2} (y - 1)}$ .

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Step 1.2.1.1

Factor $- 1$ out of $- \frac{x^{2}}{y^{2} (y - 1)}$ .

$\frac{d y}{d x} = - (\frac{x^{2}}{y^{2} (y - 1)}) - \frac{25}{y^{2} (y - 1)}$

Step 1.2.1.2

Factor $- 1$ out of $- \frac{25}{y^{2} (y - 1)}$ .

$\frac{d y}{d x} = - (\frac{x^{2}}{y^{2} (y - 1)}) - (\frac{25}{y^{2} (y - 1)})$

Step 1.2.1.3

Factor $- 1$ out of $- (\frac{x^{2}}{y^{2} (y - 1)}) - (\frac{25}{y^{2} (y - 1)})$ .

$\frac{d y}{d x} = - (\frac{x^{2}}{y^{2} (y - 1)} + \frac{25}{y^{2} (y - 1)})$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{d y}{d x} = - \frac{x^{2} + 25}{y^{2} (y - 1)}$

Step 1.3

Multiply both sides by $y^{2} (y - 1)$ .

$y^{2} (y - 1) \frac{d y}{d x} = y^{2} (y - 1) (- \frac{x^{2} + 25}{y^{2} (y - 1)})$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

$y^{2} (y - 1) \frac{d y}{d x} = - y^{2} (y - 1) \frac{x^{2} + 25}{y^{2} (y - 1)}$

Step 1.4.2

Cancel the common factor of $y^{2} (y - 1)$ .

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Step 1.4.2.1

Factor $y^{2} (y - 1)$ out of $- y^{2} (y - 1)$ .

$y^{2} (y - 1) \frac{d y}{d x} = y^{2} (y - 1) (- 1) \frac{x^{2} + 25}{y^{2} (y - 1)}$

Step 1.4.2.2

Cancel the common factor.

$y^{2} (y - 1) \frac{d y}{d x} = y^{2} (y - 1) \cdot - 1 \frac{x^{2} + 25}{y^{2} (y - 1)}$

Step 1.4.2.3

Rewrite the expression.

$y^{2} (y - 1) \frac{d y}{d x} = - (x^{2} + 25)$

Step 1.4.3

Apply the distributive property.

$y^{2} (y - 1) \frac{d y}{d x} = - x^{2} - 1 \cdot 25$

Step 1.4.4

Multiply $- 1$ by $25$ .

$y^{2} (y - 1) \frac{d y}{d x} = - x^{2} - 25$

Step 1.5

Rewrite the equation.

$y^{2} (y - 1) d y = (- x^{2} - 25) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} (y - 1) d y = \int - x^{2} - 25 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Multiply $y^{2} (y - 1)$ .

$\int y^{2} y + y^{2} \cdot - 1 d y = \int - x^{2} - 25 d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

$\int y^{2} y^{1} + y^{2} \cdot - 1 d y = \int - x^{2} - 25 d x$

Step 2.2.2.1.1.2

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

$\int y^{2 + 1} + y^{2} \cdot - 1 d y = \int - x^{2} - 25 d x$

Step 2.2.2.1.2

Add $2$ and $1$ .

$\int y^{3} + y^{2} \cdot - 1 d y = \int - x^{2} - 25 d x$

Step 2.2.2.2

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

$\int y^{3} - 1 \cdot y^{2} d y = \int - x^{2} - 25 d x$

Step 2.2.2.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$\int y^{3} - y^{2} d y = \int - x^{2} - 25 d x$

Step 2.2.3

Split the single integral into multiple integrals.

$\int y^{3} d y + \int - y^{2} d y = \int - x^{2} - 25 d x$

Step 2.2.4

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$\frac{1}{4} y^{4} + C_{1} + \int - y^{2} d y = \int - x^{2} - 25 d x$

Step 2.2.5

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

$\frac{1}{4} y^{4} + C_{1} - \int y^{2} d y = \int - x^{2} - 25 d x$

Step 2.2.6

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

$\frac{1}{4} y^{4} + C_{1} - (\frac{1}{3} y^{3} + C_{2}) = \int - x^{2} - 25 d x$

Step 2.2.7

Simplify.

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = \int - x^{2} - 25 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = \int - x^{2} d x + \int - 25 d x$

Step 2.3.2

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

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = - \int x^{2} d x + \int - 25 d x$

Step 2.3.3

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

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = - (\frac{1}{3} x^{3} + C_{4}) + \int - 25 d x$

Step 2.3.4

Apply the constant rule.

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = - (\frac{1}{3} x^{3} + C_{4}) - 25 x + C_{5}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = - (\frac{x^{3}}{3} + C_{4}) - 25 x + C_{5}$

Step 2.3.5.2

Simplify.

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} + C_{3} = - \frac{1}{3} x^{3} - 25 x + C_{6}$

Step 2.4

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

$\frac{1}{4} y^{4} - \frac{1}{3} y^{3} = - \frac{1}{3} x^{3} - 25 x + K$