Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Combine.

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

Step 1.2.2

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

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Multiply $4$ by $1$ .

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

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

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

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

$\int y^{- 2} d y = \int \frac{4}{x} d x$

Step 2.2.2

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

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

Step 2.2.3

Rewrite $- y^{- 1} + C_{1}$ as $- \frac{1}{y} + C_{1}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

$- \frac{1}{y} + C_{1} = 4 (\ln (| x |) + C_{2})$

Step 2.3.3

Simplify.

$- \frac{1}{y} + C_{1} = 4 \ln (| x |) + C_{2}$

Step 2.4

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

$- \frac{1}{y} = 4 \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Simplify $4 \ln (| x |)$ by moving $4$ inside the logarithm.

$- \frac{1}{y} = \ln ({| x |}^{4}) + C$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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 3.2.2

The LCM of one and any expression is the expression.

$y$

Step 3.3

Multiply each term in $- \frac{1}{y} = \ln ({| x |}^{4}) + C$ by $y$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{1}{y} = \ln ({| x |}^{4}) + C$ by $y$ .

$- \frac{1}{y} y = \ln ({| x |}^{4}) y + C y$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

$\frac{- 1}{y} y = \ln ({| x |}^{4}) y + C y$

Step 3.3.2.1.2

Cancel the common factor.

$\frac{- 1}{y} y = \ln ({| x |}^{4}) y + C y$

Step 3.3.2.1.3

Rewrite the expression.

$- 1 = \ln ({| x |}^{4}) y + C y$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$- 1 = \ln (x^{4}) y + C y$

Step 3.3.3.2

Reorder factors in $\ln (x^{4}) y + C y$ .

$- 1 = y \ln (x^{4}) + C y$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $y \ln (x^{4}) + C y = - 1$ .

$y \ln (x^{4}) + C y = - 1$

Step 3.4.2

Factor $y$ out of $y \ln (x^{4}) + C y$ .

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Step 3.4.2.1

Factor $y$ out of $y \ln (x^{4})$ .

$y (\ln (x^{4})) + C y = - 1$

Step 3.4.2.2

Factor $y$ out of $C y$ .

$y (\ln (x^{4})) + y C = - 1$

Step 3.4.2.3

Factor $y$ out of $y (\ln (x^{4})) + y C$ .

$y (\ln (x^{4}) + C) = - 1$

Step 3.4.3

Divide each term in $y (\ln (x^{4}) + C) = - 1$ by $\ln (x^{4}) + C$ and simplify.

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Step 3.4.3.1

Divide each term in $y (\ln (x^{4}) + C) = - 1$ by $\ln (x^{4}) + C$ .

$\frac{y (\ln (x^{4}) + C)}{\ln (x^{4}) + C} = \frac{- 1}{\ln (x^{4}) + C}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $\ln (x^{4}) + C$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{y (\ln (x^{4}) + C)}{\ln (x^{4}) + C} = \frac{- 1}{\ln (x^{4}) + C}$

Step 3.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 1}{\ln (x^{4}) + C}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{\ln (x^{4}) + C}$