Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

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

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

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

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.

$y, 1, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y$

Step 3.2

Multiply each term in $- \frac{1}{y} = x + e^{x} + K$ by $y$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{y} = x + e^{x} + K$ by $y$ .

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

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.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Reorder factors in $x y + e^{x} y + K y$ .

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

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $x y + y e^{x} + K y = - 1$ .

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

Step 3.3.2

Factor $y$ out of $x y + y e^{x} + K y$ .

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Step 3.3.2.1

Factor $y$ out of $x y$ .

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

Step 3.3.2.2

Factor $y$ out of $y e^{x}$ .

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

Step 3.3.2.3

Factor $y$ out of $K y$ .

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

Step 3.3.2.4

Factor $y$ out of $y x + y (e^{x})$ .

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

Step 3.3.2.5

Factor $y$ out of $y (x + e^{x}) + y K$ .

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

Step 3.3.3

Divide each term in $y (x + e^{x} + K) = - 1$ by $x + e^{x} + K$ and simplify.

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Step 3.3.3.1

Divide each term in $y (x + e^{x} + K) = - 1$ by $x + e^{x} + K$ .

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $x + e^{x} + K$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Divide $y$ by $1$ .

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

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.

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