Calculus Examples

$\frac{e^{2 x}}{e^{3 y}} d x + \frac{e^{2 y}}{e^{3 x}} d y = 0$

Step 1

Subtract $\frac{e^{2 x}}{e^{3 y}} d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $e^{3 x + 3 y}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $e^{2 y}$ and $e^{3 x}$ .

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Step 3.1.1

Factor $e^{3 x}$ out of $e^{2 y}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Multiply by $1$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.2.4

Divide $e^{2 y - 3 x}$ by $1$ .

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

Step 3.2

Multiply $e^{3 x + 3 y}$ by $e^{2 y - 3 x}$ by adding the exponents.

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Step 3.2.1

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

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

Step 3.2.2

Combine the opposite terms in $3 x + 3 y + 2 y - 3 x$ .

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Step 3.2.2.1

Subtract $3 x$ from $3 x$ .

$e^{3 y + 2 y + 0} d y = e^{3 x + 3 y} (- \frac{e^{2 x}}{e^{3 y}}) d x$

Step 3.2.2.2

Add $3 y + 2 y$ and $0$ .

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

Step 3.2.3

Add $3 y$ and $2 y$ .

$e^{5 y} d y = e^{3 x + 3 y} (- \frac{e^{2 x}}{e^{3 y}}) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{5 y} d y = - e^{3 x + 3 y} \frac{e^{2 x}}{e^{3 y}} d x$

Step 3.4

Cancel the common factor of $e^{2 x}$ and $e^{3 y}$ .

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Step 3.4.1

Factor $e^{3 y}$ out of $e^{2 x}$ .

$e^{5 y} d y = - e^{3 x + 3 y} \frac{e^{3 y} e^{2 x - 3 y}}{e^{3 y}} d x$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Multiply by $1$ .

$e^{5 y} d y = - e^{3 x + 3 y} \frac{e^{3 y} e^{2 x - 3 y}}{e^{3 y} \cdot 1} d x$

Step 3.4.2.2

Cancel the common factor.

$e^{5 y} d y = - e^{3 x + 3 y} \frac{e^{3 y} e^{2 x - 3 y}}{e^{3 y} \cdot 1} d x$

Step 3.4.2.3

Rewrite the expression.

$e^{5 y} d y = - e^{3 x + 3 y} \frac{e^{2 x - 3 y}}{1} d x$

Step 3.4.2.4

Divide $e^{2 x - 3 y}$ by $1$ .

$e^{5 y} d y = - e^{3 x + 3 y} e^{2 x - 3 y} d x$

Step 3.5

Multiply $e^{3 x + 3 y}$ by $e^{2 x - 3 y}$ by adding the exponents.

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Step 3.5.1

Move $e^{2 x - 3 y}$ .

$e^{5 y} d y = - (e^{2 x - 3 y} e^{3 x + 3 y}) d x$

Step 3.5.2

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

$e^{5 y} d y = - e^{2 x - 3 y + 3 x + 3 y} d x$

Step 3.5.3

Combine the opposite terms in $2 x - 3 y + 3 x + 3 y$ .

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Step 3.5.3.1

Add $- 3 y$ and $3 y$ .

$e^{5 y} d y = - e^{2 x + 3 x + 0} d x$

Step 3.5.3.2

Add $2 x + 3 x$ and $0$ .

$e^{5 y} d y = - e^{2 x + 3 x} d x$

Step 3.5.4

Add $2 x$ and $3 x$ .

$e^{5 y} d y = - e^{5 x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int e^{5 y} d y = \int - e^{5 x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Let $u_{1} = 5 y$ . Then $d u_{1} = 5 d y$ , so $\frac{1}{5} d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.2.1.1

Let $u_{1} = 5 y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 4.2.1.1.1

Differentiate $5 y$ .

$\frac{d}{d y} [5 y]$

Step 4.2.1.1.2

Since $5$ is constant with respect to $y$ , the derivative of $5 y$ with respect to $y$ is $5 \frac{d}{d y} [y]$ .

$5 \frac{d}{d y} [y]$

Step 4.2.1.1.3

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

$5 \cdot 1$

Step 4.2.1.1.4

Multiply $5$ by $1$ .

$5$

Step 4.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int e^{u_{1}} \frac{1}{5} d u_{1} = \int - e^{5 x} d x$

Step 4.2.2

Combine $e^{u_{1}}$ and $\frac{1}{5}$ .

$\int \frac{e^{u_{1}}}{5} d u_{1} = \int - e^{5 x} d x$

Step 4.2.3

Since $\frac{1}{5}$ is constant with respect to $u_{1}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int e^{u_{1}} d u_{1} = \int - e^{5 x} d x$

Step 4.2.4

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$\frac{1}{5} (e^{u_{1}} + C_{1}) = \int - e^{5 x} d x$

Step 4.2.5

Simplify.

$\frac{1}{5} e^{u_{1}} + C_{1} = \int - e^{5 x} d x$

Step 4.2.6

Replace all occurrences of $u_{1}$ with $5 y$ .

$\frac{1}{5} e^{5 y} + C_{1} = \int - e^{5 x} d x$

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.

$\frac{1}{5} e^{5 y} + C_{1} = - \int e^{5 x} d x$

Step 4.3.2

Let $u_{2} = 5 x$ . Then $d u_{2} = 5 d x$ , so $\frac{1}{5} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.3.2.1

Let $u_{2} = 5 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 4.3.2.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 4.3.2.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 4.3.2.1.3

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

$5 \cdot 1$

Step 4.3.2.1.4

Multiply $5$ by $1$ .

$5$

Step 4.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{5} e^{5 y} + C_{1} = - \int e^{u_{2}} \frac{1}{5} d u_{2}$

Step 4.3.3

Combine $e^{u_{2}}$ and $\frac{1}{5}$ .

$\frac{1}{5} e^{5 y} + C_{1} = - \int \frac{e^{u_{2}}}{5} d u_{2}$

Step 4.3.4

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} e^{5 y} + C_{1} = - (\frac{1}{5} \int e^{u_{2}} d u_{2})$

Step 4.3.5

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{5} e^{5 y} + C_{1} = - \frac{1}{5} (e^{u_{2}} + C_{2})$

Step 4.3.6

Simplify.

$\frac{1}{5} e^{5 y} + C_{1} = - \frac{1}{5} e^{u_{2}} + C_{2}$

Step 4.3.7

Replace all occurrences of $u_{2}$ with $5 x$ .

$\frac{1}{5} e^{5 y} + C_{1} = - \frac{1}{5} e^{5 x} + C_{2}$

Step 4.4

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

$\frac{1}{5} e^{5 y} = - \frac{1}{5} e^{5 x} + K$

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} e^{5 y}) = 5 (- \frac{1}{5} e^{5 x} + K)$

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 $5 (\frac{1}{5} e^{5 y})$ .

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Step 5.2.1.1.1

Combine $\frac{1}{5}$ and $e^{5 y}$ .

$5 \frac{e^{5 y}}{5} = 5 (- \frac{1}{5} e^{5 x} + K)$

Step 5.2.1.1.2

Cancel the common factor of $5$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$5 \frac{e^{5 y}}{5} = 5 (- \frac{1}{5} e^{5 x} + K)$

Step 5.2.1.1.2.2

Rewrite the expression.

$e^{5 y} = 5 (- \frac{1}{5} e^{5 x} + K)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $5 (- \frac{1}{5} e^{5 x} + K)$ .

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Step 5.2.2.1.1

Combine $e^{5 x}$ and $\frac{1}{5}$ .

$e^{5 y} = 5 (- \frac{e^{5 x}}{5} + K)$

Step 5.2.2.1.2

Apply the distributive property.

$e^{5 y} = 5 (- \frac{e^{5 x}}{5}) + 5 K$

Step 5.2.2.1.3

Cancel the common factor of $5$ .

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Step 5.2.2.1.3.1

Move the leading negative in $- \frac{e^{5 x}}{5}$ into the numerator.

$e^{5 y} = 5 \frac{- e^{5 x}}{5} + 5 K$

Step 5.2.2.1.3.2

Cancel the common factor.

$e^{5 y} = 5 \frac{- e^{5 x}}{5} + 5 K$

Step 5.2.2.1.3.3

Rewrite the expression.

$e^{5 y} = - e^{5 x} + 5 K$

Step 5.3

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

$\ln (e^{5 y}) = \ln (- e^{5 x} + 5 K)$

Step 5.4

Expand the left side.

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Step 5.4.1

Expand $\ln (e^{5 y})$ by moving $5 y$ outside the logarithm.

$5 y \ln (e) = \ln (- e^{5 x} + 5 K)$

Step 5.4.2

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

$5 y \cdot 1 = \ln (- e^{5 x} + 5 K)$

Step 5.4.3

Multiply $5$ by $1$ .

$5 y = \ln (- e^{5 x} + 5 K)$

Step 5.5

Divide each term in $5 y = \ln (- e^{5 x} + 5 K)$ by $5$ and simplify.

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Step 5.5.1

Divide each term in $5 y = \ln (- e^{5 x} + 5 K)$ by $5$ .

$\frac{5 y}{5} = \frac{\ln (- e^{5 x} + 5 K)}{5}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{\ln (- e^{5 x} + 5 K)}{5}$

Step 5.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (- e^{5 x} + 5 K)}{5}$

Step 6

Simplify the constant of integration.

$y = \frac{\ln (- e^{5 x} + K)}{5}$