Calculus Examples

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

Step 1

Subtract $x d x$ from both sides of the equation.

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

Step 2

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

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

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 $y e^{- x}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

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.

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

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Negate the exponent of $e^{- x}$ and move it out of the denominator.

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

Step 4.3.2.2

Multiply the exponents in ${(e^{- x})}^{- 1}$ .

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Multiply $- x \cdot - 1$ .

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Step 4.3.2.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.2.2.2

Multiply $x$ by $1$ .

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

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 $2 (\frac{1}{2} y^{2})$ .

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- (x e^{x} - e^{x}) + K)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 (- (x e^{x} - e^{x}) + K)$ .

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Step 5.2.2.1.1

Simplify each term.

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Step 5.2.2.1.1.1

Apply the distributive property.

$y^{2} = 2 (- (x e^{x}) - - e^{x} + K)$

Step 5.2.2.1.1.2

Multiply $- - e^{x}$ .

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Step 5.2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$y^{2} = 2 (- x e^{x} + 1 e^{x} + K)$

Step 5.2.2.1.1.2.2

Multiply $e^{x}$ by $1$ .

$y^{2} = 2 (- x e^{x} + e^{x} + K)$

Step 5.2.2.1.2

Simplify by multiplying through.

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Step 5.2.2.1.2.1

Apply the distributive property.

$y^{2} = 2 (- x e^{x}) + 2 e^{x} + 2 K$

Step 5.2.2.1.2.2

Multiply $- 1$ by $2$ .

$y^{2} = - 2 x e^{x} + 2 e^{x} + 2 K$

Step 5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- 2 x e^{x} + 2 e^{x} + 2 K}$

Step 5.4

Factor $2$ out of $- 2 x e^{x} + 2 e^{x} + 2 K$ .

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Step 5.4.1

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

$y = \pm \sqrt{2 (- x e^{x}) + 2 e^{x} + 2 K}$

Step 5.4.2

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

$y = \pm \sqrt{2 (- x e^{x}) + 2 (e^{x}) + 2 K}$

Step 5.4.3

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (- x e^{x}) + 2 (e^{x}) + 2 K}$

Step 5.4.4

Factor $2$ out of $2 (- x e^{x}) + 2 (e^{x})$ .

$y = \pm \sqrt{2 (- x e^{x} + e^{x}) + 2 K}$

Step 5.4.5

Factor $2$ out of $2 (- x e^{x} + e^{x}) + 2 K$ .

$y = \pm \sqrt{2 (- x e^{x} + e^{x} + K)}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (- x e^{x} + e^{x} + K)}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (- x e^{x} + e^{x} + K)}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (- x e^{x} + e^{x} + K)}$

$y = - \sqrt{2 (- x e^{x} + e^{x} + K)}$

$y = \sqrt{2 (- x e^{x} + e^{x} + K)}$

$y = - \sqrt{2 (- x e^{x} + e^{x} + K)}$

$y = \sqrt{2 (- x e^{x} + e^{x} + K)}$

$y = - \sqrt{2 (- x e^{x} + e^{x} + K)}$