Calculus Examples

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

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

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

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

Step 1.1.2

Subtract $y$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $\frac{1}{y^{2}} - y$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Simplify the denominator.

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

Combine the numerators over the common denominator.

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

Step 2.2.1.1.4

Simplify the numerator.

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Step 2.2.1.1.4.1

Rewrite $1$ as $1^{3}$ .

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

Step 2.2.1.1.4.2

Rewrite $y \cdot y^{2}$ as $y^{3}$ .

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Step 2.2.1.1.4.2.1

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

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Step 2.2.1.1.4.2.1.1

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

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

Step 2.2.1.1.4.2.1.2

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

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

Step 2.2.1.1.4.2.2

Add $1$ and $2$ .

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

Step 2.2.1.1.4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = y$ .

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

Step 2.2.1.1.4.4

Simplify.

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Step 2.2.1.1.4.4.1

One to any power is one.

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

Step 2.2.1.1.4.4.2

Multiply $y$ by $1$ .

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

Step 2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.1.3

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

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

Step 2.2.2

Let $u = (1 - y) (1 + y + y^{2})$ . Then $d u = - 3 y^{2} d y$ , so $- \frac{1}{3} d u = y^{2} d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = (1 - y) (1 + y + y^{2})$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = 1 - y$ and $g (y) = 1 + y + y^{2}$ .

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

Step 2.2.2.1.3

Differentiate.

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Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$(1 - y) (0 + \frac{d}{d y} [y] + \frac{d}{d y} [y^{2}]) + (1 + y + y^{2}) \frac{d}{d y} [1 - y]$

Step 2.2.2.1.3.3

Add $0$ and $\frac{d}{d y} [y]$ .

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

Step 2.2.2.1.3.4

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

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

Step 2.2.2.1.3.5

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

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

Step 2.2.2.1.3.6

By the Sum Rule, the derivative of $1 - y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- y]$ .

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

Step 2.2.2.1.3.7

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$(1 - y) (1 + 2 y) + (1 + y + y^{2}) (0 + \frac{d}{d y} [- y])$

Step 2.2.2.1.3.8

Add $0$ and $\frac{d}{d y} [- y]$ .

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

Step 2.2.2.1.3.9

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

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

Step 2.2.2.1.3.10

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

$(1 - y) (1 + 2 y) + (1 + y + y^{2}) (- 1 \cdot 1)$

Step 2.2.2.1.3.11

Simplify the expression.

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Step 2.2.2.1.3.11.1

Multiply $- 1$ by $1$ .

$(1 - y) (1 + 2 y) + (1 + y + y^{2}) \cdot - 1$

Step 2.2.2.1.3.11.2

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

$(1 - y) (1 + 2 y) - 1 \cdot (1 + y + y^{2})$

Step 2.2.2.1.3.11.3

Rewrite $- 1 (1 + y + y^{2})$ as $- (1 + y + y^{2})$ .

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

Step 2.2.2.1.4

Simplify.

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Step 2.2.2.1.4.1

Apply the distributive property.

$(1 - y) (1 + 2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.2

Apply the distributive property.

$(1 - y) \cdot 1 + (1 - y) (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.3

Apply the distributive property.

$1 \cdot 1 - y \cdot 1 + (1 - y) (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.4

Apply the distributive property.

$1 \cdot 1 - y \cdot 1 + 1 (2 y) - y (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5

Combine terms.

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Step 2.2.2.1.4.5.1

Multiply $1$ by $1$ .

$1 - y \cdot 1 + 1 (2 y) - y (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.2

Multiply $- 1$ by $1$ .

$1 - y + 1 (2 y) - y (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.3

Multiply $2$ by $1$ .

$1 - y + 2 y - y (2 y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.4

Multiply $2$ by $- 1$ .

$1 - y + 2 y - 2 y \cdot y - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.5

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

$1 - y + 2 y - 2 (y^{1} y) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.6

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

$1 - y + 2 y - 2 (y^{1} y^{1}) - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.7

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

$1 - y + 2 y - 2 y^{1 + 1} - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.8

Add $1$ and $1$ .

$1 - y + 2 y - 2 y^{2} - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.9

Add $- y$ and $2 y$ .

$1 + y - 2 y^{2} - 1 \cdot 1 - y - y^{2}$

Step 2.2.2.1.4.5.10

Multiply $- 1$ by $1$ .

$1 + y - 2 y^{2} - 1 - y - y^{2}$

Step 2.2.2.1.4.5.11

Subtract $1$ from $1$ .

$y - 2 y^{2} + 0 - y - y^{2}$

Step 2.2.2.1.4.5.12

Add $y - 2 y^{2}$ and $0$ .

$y - 2 y^{2} - y - y^{2}$

Step 2.2.2.1.4.5.13

Subtract $y$ from $y$ .

$- 2 y^{2} + 0 - y^{2}$

Step 2.2.2.1.4.5.14

Add $- 2 y^{2}$ and $0$ .

$- 2 y^{2} - y^{2}$

Step 2.2.2.1.4.5.15

Subtract $y^{2}$ from $- 2 y^{2}$ .

$- 3 y^{2}$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} \cdot \frac{1}{- 3} d u = \int d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Move the negative in front of the fraction.

$\int \frac{1}{u} (- \frac{1}{3}) d u = \int d x$

Step 2.2.3.2

Multiply $\frac{1}{u}$ by $\frac{1}{3}$ .

$\int - \frac{1}{u \cdot 3} d u = \int d x$

Step 2.2.3.3

Move $3$ to the left of $u$ .

$\int - \frac{1}{3 u} d u = \int d x$

Step 2.2.4

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

$- \int \frac{1}{3 u} d u = \int d x$

Step 2.2.5

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- (\frac{1}{3} \int \frac{1}{u} d u) = \int d x$

Step 2.2.6

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

$- \frac{1}{3} (\ln (| u |) + C_{1}) = \int d x$

Step 2.2.7

Simplify.

$- \frac{1}{3} \ln (| u |) + C_{1} = \int d x$

Step 2.2.8

Replace all occurrences of $u$ with $(1 - y) (1 + y + y^{2})$ .

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

$- \frac{1}{3} \ln (| (1 - y) (1 + y + y^{2}) |) = x + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{1}{3} \ln (| (1 - y) (1 + y + y^{2}) |)) = - 3 (x + C)$

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 $- 3 (- \frac{1}{3} \ln (| (1 - y) (1 + y + y^{2}) |))$ .

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Step 3.2.1.1.1

Expand $(1 - y) (1 + y + y^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.2.1.1.2

Simplify terms.

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Step 3.2.1.1.2.1

Simplify each term.

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Step 3.2.1.1.2.1.1

Multiply $1$ by $1$ .

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

Step 3.2.1.1.2.1.2

Multiply $y$ by $1$ .

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

Step 3.2.1.1.2.1.3

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

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

Step 3.2.1.1.2.1.4

Multiply $- 1$ by $1$ .

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

Step 3.2.1.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.1.1.2.1.5.1

Move $y$ .

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

Step 3.2.1.1.2.1.5.2

Multiply $y$ by $y$ .

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

Step 3.2.1.1.2.1.6

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

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Step 3.2.1.1.2.1.6.1

Move $y^{2}$ .

$- 3 (- \frac{1}{3} \ln (| 1 + y + y^{2} - y - y^{2} - (y^{2} y) |)) = - 3 (x + C)$

Step 3.2.1.1.2.1.6.2

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

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Step 3.2.1.1.2.1.6.2.1

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

$- 3 (- \frac{1}{3} \ln (| 1 + y + y^{2} - y - y^{2} - (y^{2} y^{1}) |)) = - 3 (x + C)$

Step 3.2.1.1.2.1.6.2.2

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

$- 3 (- \frac{1}{3} \ln (| 1 + y + y^{2} - y - y^{2} - y^{2 + 1} |)) = - 3 (x + C)$

Step 3.2.1.1.2.1.6.3

Add $2$ and $1$ .

$- 3 (- \frac{1}{3} \ln (| 1 + y + y^{2} - y - y^{2} - y^{3} |)) = - 3 (x + C)$

Step 3.2.1.1.2.2

Simplify terms.

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Step 3.2.1.1.2.2.1

Combine the opposite terms in $1 + y + y^{2} - y - y^{2} - y^{3}$ .

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Step 3.2.1.1.2.2.1.1

Subtract $y$ from $y$ .

$- 3 (- \frac{1}{3} \ln (| 1 + y^{2} + 0 - y^{2} - y^{3} |)) = - 3 (x + C)$

Step 3.2.1.1.2.2.1.2

Add $1 + y^{2}$ and $0$ .

$- 3 (- \frac{1}{3} \ln (| 1 + y^{2} - y^{2} - y^{3} |)) = - 3 (x + C)$

Step 3.2.1.1.2.2.1.3

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

$- 3 (- \frac{1}{3} \ln (| 1 + 0 - y^{3} |)) = - 3 (x + C)$

Step 3.2.1.1.2.2.1.4

Add $1$ and $0$ .

$- 3 (- \frac{1}{3} \ln (| 1 - y^{3} |)) = - 3 (x + C)$

Step 3.2.1.1.2.2.2

Combine $\ln (| 1 - y^{3} |)$ and $\frac{1}{3}$ .

$- 3 (- \frac{\ln (| 1 - y^{3} |)}{3}) = - 3 (x + C)$

Step 3.2.1.1.2.2.3

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.2.3.1

Move the leading negative in $- \frac{\ln (| 1 - y^{3} |)}{3}$ into the numerator.

$- 3 \frac{- \ln (| 1 - y^{3} |)}{3} = - 3 (x + C)$

Step 3.2.1.1.2.2.3.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- \ln (| 1 - y^{3} |)}{3} = - 3 (x + C)$

Step 3.2.1.1.2.2.3.3

Cancel the common factor.

$3 \cdot - 1 \frac{- \ln (| 1 - y^{3} |)}{3} = - 3 (x + C)$

Step 3.2.1.1.2.2.3.4

Rewrite the expression.

$- - \ln (| 1 - y^{3} |) = - 3 (x + C)$

Step 3.2.1.1.2.2.4

Multiply.

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Step 3.2.1.1.2.2.4.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| 1 - y^{3} |) = - 3 (x + C)$

Step 3.2.1.1.2.2.4.2

Multiply $\ln (| 1 - y^{3} |)$ by $1$ .

$\ln (| 1 - y^{3} |) = - 3 (x + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$\ln (| 1 - y^{3} |) = - 3 x - 3 C$

Step 3.3

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| 1 - y^{3} |)} = e^{- 3 x - 3 C}$

Step 3.4

Rewrite $\ln (| 1 - y^{3} |) = - 3 x - 3 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 3 x - 3 C} = | 1 - y^{3} |$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $| 1 - y^{3} | = e^{- 3 x - 3 C}$ .

$| 1 - y^{3} | = e^{- 3 x - 3 C}$

Step 3.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$1 - y^{3} = \pm e^{- 3 x - 3 C}$

Step 3.5.3

Subtract $1$ from both sides of the equation.

$- y^{3} = \pm e^{- 3 x - 3 C} - 1$

Step 3.5.4

Divide each term in $- y^{3} = \pm e^{- 3 x - 3 C} - 1$ by $- 1$ and simplify.

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Step 3.5.4.1

Divide each term in $- y^{3} = \pm e^{- 3 x - 3 C} - 1$ by $- 1$ .

$\frac{- y^{3}}{- 1} = \frac{\pm e^{- 3 x - 3 C}}{- 1} + \frac{- 1}{- 1}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{y^{3}}{1} = \frac{\pm e^{- 3 x - 3 C}}{- 1} + \frac{- 1}{- 1}$

Step 3.5.4.2.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{\pm e^{- 3 x - 3 C}}{- 1} + \frac{- 1}{- 1}$

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Simplify each term.

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Step 3.5.4.3.1.1

Move the negative one from the denominator of $\frac{\pm e^{- 3 x - 3 C}}{- 1}$ .

$y^{3} = - 1 \cdot (\pm e^{- 3 x - 3 C}) + \frac{- 1}{- 1}$

Step 3.5.4.3.1.2

Rewrite $- 1 \cdot (\pm e^{- 3 x - 3 C})$ as $- (\pm e^{- 3 x - 3 C})$ .

$y^{3} = - (\pm e^{- 3 x - 3 C}) + \frac{- 1}{- 1}$

Step 3.5.4.3.1.3

Divide $- 1$ by $- 1$ .

$y^{3} = - (\pm e^{- 3 x - 3 C}) + 1$

Step 3.5.5

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

$y = \sqrt[3]{- (\pm e^{- 3 x - 3 C}) + 1}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \sqrt[3]{- (\pm e^{- 3 x + C}) + 1}$

Step 4.2

Rewrite $e^{- 3 x + C}$ as $e^{- 3 x} e^{C}$ .

$y = \sqrt[3]{- (\pm e^{- 3 x} e^{C}) + 1}$

Step 4.3

Reorder $e^{- 3 x}$ and $e^{C}$ .

$y = \sqrt[3]{- (\pm e^{C} e^{- 3 x}) + 1}$

Step 4.4

Combine constants with the plus or minus.

$y = \sqrt[3]{- (C e^{- 3 x}) + 1}$