Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Apply the constant rule.

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

Step 4.4

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

$\ln (| y |) = - x + C$

Step 5

Solve for $y$ .

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Step 5.1

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

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

Step 5.2

Rewrite $\ln (| y |) = - x + C$ 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 $b^{y} = x$ .

$e^{- x + C} = | y |$

Step 5.3

Solve for $y$ .

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Step 5.3.1

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

$| y | = e^{- x + C}$

Step 5.3.2

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

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

Step 6

Group the constant terms together.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Combine constants with the plus or minus.

$y = C e^{- x}$