Calculus Examples

\frac{d y}{d x} = 9 x^{2} y^{2}

$\frac{d y}{d x} = 9 x^{2} y^{2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by

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

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{1}{y^{2}} (9 x^{2} y^{2})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{2}} \frac{d y}{d x} = 9 \frac{1}{y^{2}} (x^{2} y^{2})$

Step 1.2.2

Combine

$9$ and

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

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{9}{y^{2}} (x^{2} y^{2})$

Step 1.2.3

Cancel the common factor of

$y^{2}$ .

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Step 1.2.3.1

Factor

$y^{2}$ out of

$x^{2} y^{2}$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{9}{y^{2}} (y^{2} x^{2})$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{9}{y^{2}} (y^{2} x^{2})$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y^{2}} \frac{d y}{d x} = 9 x^{2}$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{2}} d y = 9 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{2}} d y = \int 9 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move

$y^{2}$ out of the denominator by raising it to the

$- 1$ power.

$\int {(y^{2})}^{- 1} d y = \int 9 x^{2} d x$

Step 2.2.1.2

Multiply the exponents in

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

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Step 2.2.1.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\int y^{2 \cdot - 1} d y = \int 9 x^{2} d x$

Step 2.2.1.2.2

Multiply

$2$ by

$- 1$ .

$\int y^{- 2} d y = \int 9 x^{2} d x$

Step 2.2.2

By the Power Rule, the integral of

$y^{- 2}$ with respect to

$y$ is

$- y^{- 1}$ .

$- y^{- 1} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3

Rewrite

$- y^{- 1} + C_{1}$ as

$- \frac{1}{y} + C_{1}$ .

$- \frac{1}{y} + C_{1} = \int 9 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since

$9$ is constant with respect to

$x$ , move

$9$ out of the integral.

$- \frac{1}{y} + C_{1} = 9 \int x^{2} d x$

Step 2.3.2

By the Power Rule, the integral of

$x^{2}$ with respect to

$x$ is

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

$- \frac{1}{y} + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite

$9 (\frac{1}{3} x^{3} + C_{2})$ as

$9 (\frac{1}{3}) x^{3} + C_{2}$ .

$- \frac{1}{y} + C_{1} = 9 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine

$9$ and

$\frac{1}{3}$ .

$- \frac{1}{y} + C_{1} = \frac{9}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of

$9$ and

$3$ .

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Step 2.3.3.2.2.1

Factor

$3$ out of

$9$ .

$- \frac{1}{y} + C_{1} = \frac{3 \cdot 3}{3} x^{3} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor

$3$ out of

$3$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

$- \frac{1}{y} + C_{1} = \frac{3 \cdot 3}{3 \cdot 1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$- \frac{1}{y} + C_{1} = \frac{3}{1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.4

Divide

$3$ by

$1$ .

$- \frac{1}{y} + C_{1} = 3 x^{3} + C_{2}$

Step 2.4

Group the constant of integration on the right side as

$K$ .

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

Step 3

Solve for

$y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y$

Step 3.2

Multiply each term in

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

$y$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in

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

$y$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

$y$ .

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Step 3.2.2.1.1

Move the leading negative in

$- \frac{1}{y}$ into the numerator.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = 3 x^{3} y + K y$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as

$3 x^{3} y + K y = - 1$ .

$3 x^{3} y + K y = - 1$

Step 3.3.2

Factor

$y$ out of

$3 x^{3} y + K y$ .

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Step 3.3.2.1

Factor

$y$ out of

$3 x^{3} y$ .

$y (3 x^{3}) + K y = - 1$

Step 3.3.2.2

Factor

$y$ out of

$K y$ .

$y (3 x^{3}) + y K = - 1$

Step 3.3.2.3

Factor

$y$ out of

$y (3 x^{3}) + y K$ .

$y (3 x^{3} + K) = - 1$

Step 3.3.3

Divide each term in

$y (3 x^{3} + K) = - 1$ by

$3 x^{3} + K$ and simplify.

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Step 3.3.3.1

Divide each term in

$y (3 x^{3} + K) = - 1$ by

$3 x^{3} + K$ .

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of

$3 x^{3} + K$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Divide

$y$ by

$1$ .

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

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