Calculus Examples

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

Step 1

Let $v = e^{x - y}$ . Substitute $v$ for all occurrences of $e^{x - y}$ .

$\frac{d y}{d x} = x v$

Step 2

Find $\frac{d v}{d x}$ by differentiating $e^{x - y}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x - y$ .

$\frac{d v}{d x} = \frac{d}{d u} [e^{u}] \frac{d}{d x} [x - y]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$\frac{d v}{d x} = e^{u} \frac{d}{d x} [x - y]$

Step 2.1.3

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

$\frac{d v}{d x} = e^{x - y} \frac{d}{d x} [x - y]$

Step 2.2

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

$\frac{d v}{d x} = e^{x - y} (\frac{d}{d x} [x] + \frac{d}{d x} [- y])$

Step 2.3

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

$\frac{d v}{d x} = e^{x - y} (1 + \frac{d}{d x} [- y])$

Step 2.4

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

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

Step 2.5

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 3

Substitute $v$ for $e^{x - y}$ .

$\frac{d v}{d x} = v (1 - y')$

Step 4

Substitute the derivative back in to the differential equation.

$- \frac{\frac{d v}{d x}}{v} + 1 = x v$

Step 5

Multiply each term in $- \frac{\frac{d v}{d x}}{v} + 1 = x v$ by $- v$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v} + 1 = x v$ by $- v$ .

$- \frac{\frac{d v}{d x}}{v} (- v) + 1 (- v) = x v (- v)$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of $v$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{v}$ into the numerator.

$\frac{- \frac{d v}{d x}}{v} (- v) + 1 (- v) = x v (- v)$

Step 5.2.1.1.2

Factor $v$ out of $- v$ .

$\frac{- \frac{d v}{d x}}{v} (v \cdot - 1) + 1 (- v) = x v (- v)$

Step 5.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{v} (v \cdot - 1) + 1 (- v) = x v (- v)$

Step 5.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + 1 (- v) = x v (- v)$

Step 5.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + 1 (- v) = x v (- v)$

Step 5.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

$\frac{d v}{d x} + 1 (- v) = x v (- v)$

Step 5.2.1.4

Multiply $- v$ by $1$ .

$\frac{d v}{d x} - v = x v (- v)$

Step 5.3

Simplify the right side.

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Step 5.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - v = - x v \cdot v$

Step 5.3.2

Multiply $v$ by $v$ by adding the exponents.

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Step 5.3.2.1

Move $v$ .

$\frac{d v}{d x} - v = - x (v \cdot v)$

Step 5.3.2.2

Multiply $v$ by $v$ .

$\frac{d v}{d x} - v = - x v^{2}$

Step 6

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $v^{2}$ .

$v = v^{- 1}$

Step 7

Solve the equation for $v$ .

$y = v^{- 1}$

Step 8

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- 1}$

Step 9

Take the derivative of $v^{- 1}$ with respect to $x$ .

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Step 9.1

Take the derivative of $v^{- 1}$ .

$y' = \frac{d}{d x} [v^{- 1}]$

Step 9.2

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

$y' = \frac{d}{d x} [\frac{1}{v}]$

Step 9.3

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

$y' = \frac{v \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v]}{v^{2}}$

Step 9.4

Differentiate using the Constant Rule.

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Step 9.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v \frac{d}{d x} [1] - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.2

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

$y' = \frac{v \cdot 0 - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3

Simplify the expression.

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Step 9.4.3.1

Multiply $v$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3.2

Subtract $\frac{d}{d x} [v]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v]}{v^{2}}$

Step 9.5

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$y' = - \frac{v'}{v^{2}}$

Step 10

Substitute $- \frac{v'}{v^{2}}$ for $\frac{d v}{d x}$ and $v^{- 1}$ for $v$ in the original equation $\frac{d v}{d x} - v = - x v^{2}$ .

$- \frac{v'}{v^{2}} - v^{- 1} = - x {(v^{- 1})}^{2}$

Step 11

Solve the substituted differential equation.

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Step 11.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - v^{- 1} = - x {(v^{- 1})}^{2}$ by $- v^{2}$ to eliminate the fractions.

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Step 11.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - v^{- 1} = - x {(v^{- 1})}^{2}$ by $- v^{2}$ .

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

Step 11.1.2

Simplify the left side.

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Step 11.1.2.1

Simplify each term.

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Step 11.1.2.1.1

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

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Step 11.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{v^{2}}$ into the numerator.

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

Step 11.1.2.1.1.2

Factor $v^{2}$ out of $- v^{2}$ .

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

Step 11.1.2.1.1.3

Cancel the common factor.

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

Step 11.1.2.1.1.4

Rewrite the expression.

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

Step 11.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 11.1.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

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

Step 11.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 11.1.2.1.5

Multiply $v^{- 1}$ by $v^{2}$ by adding the exponents.

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Step 11.1.2.1.5.1

Move $v^{2}$ .

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

Step 11.1.2.1.5.2

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

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

Step 11.1.2.1.5.3

Subtract $1$ from $2$ .

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

Step 11.1.2.1.6

Simplify $- 1 \cdot - 1 v^{1}$ .

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

Step 11.1.2.1.7

Multiply $- 1$ by $- 1$ .

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

Step 11.1.2.1.8

Multiply $v$ by $1$ .

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

Step 11.1.3

Simplify the right side.

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Step 11.1.3.1

Multiply the exponents in ${(v^{- 1})}^{2}$ .

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Step 11.1.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 11.1.3.1.2

Multiply $- 1$ by $2$ .

$\frac{d v}{d x} + v = - x v^{- 2} (- v^{2})$

Step 11.1.3.2

Multiply $v^{- 2}$ by $v^{2}$ by adding the exponents.

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Step 11.1.3.2.1

Move $v^{2}$ .

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

Step 11.1.3.2.2

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

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

Step 11.1.3.2.3

Subtract $2$ from $2$ .

$\frac{d v}{d x} + v = - x v^{0} \cdot - 1$

Step 11.1.3.3

Simplify $- x v^{0} \cdot - 1$ .

$\frac{d v}{d x} + v = - x \cdot - 1$

Step 11.1.3.4

Multiply $- x \cdot - 1$ .

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Step 11.1.3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d v}{d x} + v = 1 x$

Step 11.1.3.4.2

Multiply $x$ by $1$ .

$\frac{d v}{d x} + v = x$

Step 11.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 1$ .

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Step 11.2.1

Set up the integration.

$e^{\int d x}$

Step 11.2.2

Apply the constant rule.

$e^{x + C}$

Step 11.2.3

Remove the constant of integration.

$e^{x}$

Step 11.3

Multiply each term by the integrating factor $e^{x}$ .

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Step 11.3.1

Multiply each term by $e^{x}$ .

$e^{x} \frac{d v}{d x} + e^{x} v = e^{x} x$

Step 11.3.2

Reorder factors in $e^{x} \frac{d v}{d x} + e^{x} v = e^{x} x$ .

$e^{x} \frac{d v}{d x} + v e^{x} = x e^{x}$

Step 11.4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{x} v] = x e^{x}$

Step 11.5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} v] d x = \int x e^{x} d x$

Step 11.6

Integrate the left side.

$e^{x} v = \int x e^{x} d x$

Step 11.7

Integrate the right side.

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Step 11.7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$e^{x} v = x e^{x} - \int e^{x} d x$

Step 11.7.2

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

$e^{x} v = x e^{x} - (e^{x} + C)$

Step 11.7.3

Simplify.

$e^{x} v = x e^{x} - e^{x} + C$

Step 11.8

Divide each term in $e^{x} v = x e^{x} - e^{x} + C$ by $e^{x}$ and simplify.

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Step 11.8.1

Divide each term in $e^{x} v = x e^{x} - e^{x} + C$ by $e^{x}$ .

$\frac{e^{x} v}{e^{x}} = \frac{x e^{x}}{e^{x}} + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.2

Simplify the left side.

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Step 11.8.2.1

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

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Step 11.8.2.1.1

Cancel the common factor.

$\frac{e^{x} v}{e^{x}} = \frac{x e^{x}}{e^{x}} + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.2.1.2

Divide $v$ by $1$ .

$v = \frac{x e^{x}}{e^{x}} + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.3

Simplify the right side.

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Step 11.8.3.1

Simplify each term.

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Step 11.8.3.1.1

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

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Step 11.8.3.1.1.1

Cancel the common factor.

$v = \frac{x e^{x}}{e^{x}} + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.3.1.1.2

Divide $x$ by $1$ .

$v = x + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.3.1.2

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

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Step 11.8.3.1.2.1

Cancel the common factor.

$v = x + \frac{- e^{x}}{e^{x}} + \frac{C}{e^{x}}$

Step 11.8.3.1.2.2

Divide $- 1$ by $1$ .

$v = x - 1 + \frac{C}{e^{x}}$

Step 12

Substitute $v^{- 1}$ for $v$ .

$v^{- 1} = x - 1 + \frac{C}{e^{x}}$

Step 13

Replace all occurrences of $v$ with $e^{x - y}$ .

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

Step 14

Solve for $y$ .

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Step 14.1

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

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

Step 14.2

Expand the left side.

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Step 14.2.1

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

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

Step 14.2.2

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

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

Step 14.2.3

Multiply $- x + y$ by $1$ .

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

Step 14.3

Add $x$ to both sides of the equation.

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