Calculus Examples

$e^{x} e^{y} d x - e^{- 2 y} d y = 0$

Step 1

Subtract $e^{x} e^{y} d x$ from both sides of the equation.

$- e^{- 2 y} d y = - e^{x} e^{y} d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Combine $e^{- 2 y}$ and $\frac{1}{e^{y}}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Multiply by $1$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.3.2.4

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

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

$- e^{- 3 y} d y = - (e^{x - y} e^{y}) d x$

Step 3.6

Multiply $e^{x - y}$ by $e^{y}$ by adding the exponents.

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Step 3.6.1

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

$- e^{- 3 y} d y = - e^{x - y + y} d x$

Step 3.6.2

Combine the opposite terms in $x - y + y$ .

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Step 3.6.2.1

Add $- y$ and $y$ .

$- e^{- 3 y} d y = - e^{x + 0} d x$

Step 3.6.2.2

Add $x$ and $0$ .

$- e^{- 3 y} d y = - e^{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

Let $u = - 3 y$ . Find $\frac{d u}{d y}$ .

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Step 4.2.2.1.1

Differentiate $- 3 y$ .

$\frac{d}{d y} [- 3 y]$

Step 4.2.2.1.2

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3 y$ with respect to $y$ is $- 3 \frac{d}{d y} [y]$ .

$- 3 \frac{d}{d y} [y]$

Step 4.2.2.1.3

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

$- 3 \cdot 1$

Step 4.2.2.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 4.2.2.2

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

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

Step 4.2.3

Simplify.

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Step 4.2.3.1

Move the negative in front of the fraction.

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

Step 4.2.3.2

Combine $e^{u}$ and $\frac{1}{3}$ .

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

Step 4.2.4

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

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

Step 4.2.5

Simplify.

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Step 4.2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.5.2

Multiply $\int \frac{e^{u}}{3} d u$ by $1$ .

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

Step 4.2.6

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

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

Step 4.2.7

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

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

Step 4.2.8

Simplify.

$\frac{1}{3} e^{u} + C_{1} = \int - e^{x} d x$

Step 4.2.9

Replace all occurrences of $u$ with $- 3 y$ .

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $3$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $3 (\frac{1}{3} e^{- 3 y})$ .

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Step 5.2.1.1.1

Combine $\frac{1}{3}$ and $e^{- 3 y}$ .

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

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $3 (- e^{x} + K)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Multiply $- 1$ by $3$ .

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

Step 5.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 3 y}) = \ln (- 3 e^{x} + 3 K)$

Step 5.4

Expand the left side.

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Step 5.4.1

Expand $\ln (e^{- 3 y})$ by moving $- 3 y$ outside the logarithm.

$- 3 y \ln (e) = \ln (- 3 e^{x} + 3 K)$

Step 5.4.2

The natural logarithm of $e$ is $1$ .

$- 3 y \cdot 1 = \ln (- 3 e^{x} + 3 K)$

Step 5.4.3

Multiply $- 3$ by $1$ .

$- 3 y = \ln (- 3 e^{x} + 3 K)$

Step 5.5

Divide each term in $- 3 y = \ln (- 3 e^{x} + 3 K)$ by $- 3$ and simplify.

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Step 5.5.1

Divide each term in $- 3 y = \ln (- 3 e^{x} + 3 K)$ by $- 3$ .

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 3$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 6

Simplify the constant of integration.

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