Calculus Examples

$2 \frac{d y}{d x} = 4 x e^{- x}$ , $y (0) = 5$

Step 1

Rewrite the equation.

$2 d y = 4 x e^{- x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 d y = \int 4 x e^{- x} d x$

Step 2.2

Apply the constant rule.

$2 y + C_{1} = \int 4 x e^{- x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$2 y + C_{1} = 4 \int x e^{- x} d x$

Step 2.3.2

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

$2 y + C_{1} = 4 (x (- e^{- x}) - \int - e^{- x} d x)$

Step 2.3.3

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

$2 y + C_{1} = 4 (x (- e^{- x}) - - \int e^{- x} d x)$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

$2 y + C_{1} = 4 (x (- e^{- x}) + 1 \int e^{- x} d x)$

Step 2.3.4.2

Multiply $\int e^{- x} d x$ by $1$ .

$2 y + C_{1} = 4 (x (- e^{- x}) + \int e^{- x} d x)$

Step 2.3.5

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.5.1

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

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Step 2.3.5.1.1

Differentiate $- x$ .

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

Step 2.3.5.1.2

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

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

Step 2.3.5.1.3

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

$- 1 \cdot 1$

Step 2.3.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.5.2

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

$2 y + C_{1} = 4 (x (- e^{- x}) + \int - e^{u} d u)$

Step 2.3.6

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

$2 y + C_{1} = 4 (x (- e^{- x}) - \int e^{u} d u)$

Step 2.3.7

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

$2 y + C_{1} = 4 (x (- e^{- x}) - (e^{u} + C_{2}))$

Step 2.3.8

Rewrite $4 (x (- e^{- x}) - (e^{u} + C_{2}))$ as $4 (- x e^{- x} - e^{u}) + C_{2}$ .

$2 y + C_{1} = 4 (- x e^{- x} - e^{u}) + C_{2}$

Step 2.3.9

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

$2 y + C_{1} = 4 (- x e^{- x} - e^{- x}) + C_{2}$

Step 2.4

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

$2 y = 4 (- x e^{- x} - e^{- x}) + K$

Step 3

Divide each term in $2 y = 4 (- x e^{- x} - e^{- x}) + K$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 y = 4 (- x e^{- x} - e^{- x}) + K$ by $2$ .

$\frac{2 y}{2} = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$y = \frac{4 (- x e^{- x} - e^{- x}) + K}{2}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Apply the distributive property.

$y = \frac{4 (- x e^{- x}) + 4 (- e^{- x}) + K}{2}$

Step 3.3.2.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 (x e^{- x}) + 4 (- e^{- x}) + K}{2}$

Step 3.3.2.3

Multiply $- 1$ by $4$ .

$y = \frac{- 4 x e^{- x} - 4 e^{- x} + K}{2}$

Step 3.3.3

Simplify with factoring out.

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Step 3.3.3.1

Factor $- 1$ out of $- 4 x e^{- x}$ .

$y = \frac{- (4 x e^{- x}) - 4 e^{- x} + K}{2}$

Step 3.3.3.2

Factor $- 1$ out of $- 4 e^{- x}$ .

$y = \frac{- (4 x e^{- x}) - (4 e^{- x}) + K}{2}$

Step 3.3.3.3

Factor $- 1$ out of $- (4 x e^{- x}) - (4 e^{- x})$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x}) + K}{2}$

Step 3.3.3.4

Factor $- 1$ out of $K$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x}) - 1 (- K)}{2}$

Step 3.3.3.5

Factor $- 1$ out of $- (4 x e^{- x} + 4 e^{- x}) - 1 (- K)$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x} - K)}{2}$

Step 3.3.3.6

Simplify the expression.

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Step 3.3.3.6.1

Rewrite $- (4 x e^{- x} + 4 e^{- x} - K)$ as $- 1 (4 x e^{- x} + 4 e^{- x} - K)$ .

$y = \frac{- 1 (4 x e^{- x} + 4 e^{- x} - K)}{2}$

Step 3.3.3.6.2

Move the negative in front of the fraction.

$y = - \frac{4 x e^{- x} + 4 e^{- x} - K}{2}$

Step 4

Simplify the constant of integration.

$y = - \frac{4 x e^{- x} + 4 e^{- x} + K}{2}$

Step 5

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $5$ for $y$ in $y = - \frac{4 x e^{- x} + 4 e^{- x} + K}{2}$ .

$5 = - \frac{4 \cdot 0 e^{- 0} + 4 e^{0} + K}{2}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $- \frac{4 \cdot (0 e^{- 0}) + 4 e^{0} + K}{2} = 5$ .

$- \frac{4 \cdot (0 e^{- 0}) + 4 e^{0} + K}{2} = 5$

Step 6.2

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{4 \cdot (0 e^{- 0}) + 4 e^{0} + K}{2}) = - 2 \cdot 5$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $- 2 (- \frac{4 \cdot (0 e^{- 0}) + 4 e^{0} + K}{2})$ .

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Step 6.3.1.1.1

Reduce the expression by cancelling the common factors.

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Step 6.3.1.1.1.1

Cancel the common factor of $2$ .

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Step 6.3.1.1.1.1.1

Move the leading negative in $- \frac{4 \cdot (0 e^{- 0}) + 4 e^{0} + K}{2}$ into the numerator.

$- 2 \frac{- (4 \cdot (0 e^{- 0}) + 4 e^{0} + K)}{2} = - 2 \cdot 5$

Step 6.3.1.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- (4 \cdot (0 e^{- 0}) + 4 e^{0} + K)}{2} = - 2 \cdot 5$

Step 6.3.1.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- (4 \cdot (0 e^{- 0}) + 4 e^{0} + K)}{2} = - 2 \cdot 5$

Step 6.3.1.1.1.1.4

Rewrite the expression.

$- - (4 \cdot (0 e^{- 0}) + 4 e^{0} + K) = - 2 \cdot 5$

Step 6.3.1.1.1.2

Simplify the expression.

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Step 6.3.1.1.1.2.1

Multiply $- 1$ by $0$ .

$- - (4 \cdot (0 e^{0}) + 4 e^{0} + K) = - 2 \cdot 5$

Step 6.3.1.1.1.2.2

Multiply by zero.

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Step 6.3.1.1.1.2.2.1

Multiply $0$ by $e^{0}$ .

$- - (4 \cdot 0 + 4 e^{0} + K) = - 2 \cdot 5$

Step 6.3.1.1.1.2.2.2

Multiply $4$ by $0$ .

$- - (0 + 4 e^{0} + K) = - 2 \cdot 5$

Step 6.3.1.1.1.2.3

Multiply $- 1$ by $- 1$ .

$1 (0 + 4 e^{0} + K) = - 2 \cdot 5$

Step 6.3.1.1.1.2.4

Multiply $0 + 4 e^{0} + K$ by $1$ .

$0 + 4 e^{0} + K = - 2 \cdot 5$

Step 6.3.1.1.1.2.5

Add $0$ and $4 e^{0}$ .

$4 e^{0} + K = - 2 \cdot 5$

Step 6.3.1.1.2

Simplify each term.

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Step 6.3.1.1.2.1

Anything raised to $0$ is $1$ .

$4 \cdot 1 + K = - 2 \cdot 5$

Step 6.3.1.1.2.2

Multiply $4$ by $1$ .

$4 + K = - 2 \cdot 5$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $- 2$ by $5$ .

$4 + K = - 10$

Step 6.4

Move all terms not containing $K$ to the right side of the equation.

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Step 6.4.1

Subtract $4$ from both sides of the equation.

$K = - 10 - 4$

Step 6.4.2

Subtract $4$ from $- 10$ .

$K = - 14$

Step 7

Substitute $- 14$ for $K$ in $y = - \frac{4 x e^{- x} + 4 e^{- x} + K}{2}$ and simplify.

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Step 7.1

Substitute $- 14$ for $K$ .

$y = - \frac{4 x e^{- x} + 4 e^{- x} - 14}{2}$

Step 7.2

Cancel the common factor of $4 x e^{- x} + 4 e^{- x} - 14$ and $2$ .

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Step 7.2.1

Factor $2$ out of $4 x e^{- x}$ .

$y = - \frac{2 (2 x e^{- x}) + 4 e^{- x} - 14}{2}$

Step 7.2.2

Factor $2$ out of $4 e^{- x}$ .

$y = - \frac{2 (2 x e^{- x}) + 2 (2 e^{- x}) - 14}{2}$

Step 7.2.3

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

$y = - \frac{2 (2 x e^{- x} + 2 e^{- x}) - 14}{2}$

Step 7.2.4

Factor $2$ out of $- 14$ .

$y = - \frac{2 (2 x e^{- x} + 2 e^{- x}) + 2 \cdot - 7}{2}$

Step 7.2.5

Factor $2$ out of $2 (2 x e^{- x} + 2 e^{- x}) + 2 (- 7)$ .

$y = - \frac{2 (2 x e^{- x} + 2 e^{- x} - 7)}{2}$

Step 7.2.6

Cancel the common factors.

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Step 7.2.6.1

Factor $2$ out of $2$ .

$y = - \frac{2 (2 x e^{- x} + 2 e^{- x} - 7)}{2 (1)}$

Step 7.2.6.2

Cancel the common factor.

$y = - \frac{2 (2 x e^{- x} + 2 e^{- x} - 7)}{2 \cdot 1}$

Step 7.2.6.3

Rewrite the expression.

$y = - \frac{2 x e^{- x} + 2 e^{- x} - 7}{1}$

Step 7.2.6.4

Divide $2 x e^{- x} + 2 e^{- x} - 7$ by $1$ .

$y = - (2 x e^{- x} + 2 e^{- x} - 7)$

Step 7.3

Apply the distributive property.

$y = - (2 x e^{- x}) - (2 e^{- x}) - - 7$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $2$ by $- 1$ .

$y = - 2 (x e^{- x}) - (2 e^{- x}) - - 7$

Step 7.4.2

Multiply $2$ by $- 1$ .

$y = - 2 (x e^{- x}) - 2 e^{- x} - - 7$

Step 7.4.3

Multiply $- 1$ by $- 7$ .

$y = - 2 (x e^{- x}) - 2 e^{- x} + 7$

$y = - 2 x e^{- x} - 2 e^{- x} + 7$