Calculus Examples

x e^{- t} \frac{d x}{d t} = t

$x e^{- t} \frac{d x}{d t} = t$ ,

x (0) = 1

$x (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Divide each term in

x e^{- t} \frac{d x}{d t} = t

$x e^{- t} \frac{d x}{d t} = t$ by

x e^{- t}

$x e^{- t}$ and simplify.

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Step 1.1.1

Divide each term in

x e^{- t} \frac{d x}{d t} = t

$x e^{- t} \frac{d x}{d t} = t$ by

x e^{- t}

$x e^{- t}$ .

\frac{x e^{- t} \frac{d x}{d t}}{x e^{- t}} = \frac{t}{x e^{- t}}

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of

x

$x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of

$e^{- t}$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Divide

$\frac{d x}{d t}$ by

$1$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by

$x$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Combine.

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

Step 1.4.2

Cancel the common factor of

$x$ .

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Step 1.4.2.1

Factor

$x$ out of

$e^{- t} x$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

Multiply

$t$ by

$1$ .

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

By the Power Rule, the integral of

$x$ with respect to

$x$ is

$\frac{1}{2} x^{2}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

Negate the exponent of

$e^{- t}$ and move it out of the denominator.

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

Step 2.3.1.2

Multiply the exponents in

${(e^{- t})}^{- 1}$ .

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Step 2.3.1.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.2.2

Multiply

$- t \cdot - 1$ .

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Step 2.3.1.2.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 2.3.1.2.2.2

Multiply

$t$ by

$1$ .

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

Step 2.3.2

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = t$ and

$d v = e^{t}$ .

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

Step 2.3.3

The integral of

$e^{t}$ with respect to

$t$ is

$e^{t}$ .

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

Group the constant of integration on the right side as

$K$ .

$\frac{1}{2} x^{2} = e^{t} t - e^{t} + K$

Step 3

Solve for

$x$ .

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Step 3.1

Multiply both sides of the equation by

$2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify

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

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Step 3.2.1.1.1

Combine

$\frac{1}{2}$ and

$x^{2}$ .

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

Step 3.2.1.1.2

Cancel the common factor of

$2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

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

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

Multiply

$- 1$ by

$2$ .

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

Step 3.2.2.1.3

Reorder factors in

$2 e^{t} t - 2 e^{t} + 2 K$ .

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

Step 3.3

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

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

Step 3.4

Factor

$2$ out of

$2 t e^{t} - 2 e^{t} + 2 K$ .

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Step 3.4.1

Factor

$2$ out of

$2 t e^{t}$ .

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

Step 3.4.2

Factor

$2$ out of

$- 2 e^{t}$ .

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

Step 3.4.3

Factor

$2$ out of

$2 K$ .

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

Step 3.4.4

Factor

$2$ out of

$2 (t e^{t}) + 2 (- e^{t})$ .

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

Step 3.4.5

Factor

$2$ out of

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

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

Step 3.5

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

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Step 3.5.1

First, use the positive value of the

$\pm$ to find the first solution.

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

Step 3.5.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 3.5.3

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

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

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

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

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

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

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

Step 4

Since

$x$ is positive in the initial condition

$(0, 1)$ , only consider

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$ to find the

$K$ . Substitute

$0$ for

$t$ and

$1$ for

$x$ .

$1 = \sqrt{2 (0 e^{0} - e^{0} + K)}$

Step 5

Solve for

$K$ .

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Step 5.1

Rewrite the equation as

$\sqrt{2 (0 e^{0} - e^{0} + K)} = 1$ .

$\sqrt{2 (0 e^{0} - e^{0} + K)} = 1$

Step 5.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 (0 e^{0} - e^{0} + K)}}^{2} = 1^{2}$

Step 5.3

Simplify each side of the equation.

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Step 5.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2 (0 e^{0} - e^{0} + K)}$ as

${(2 (0 e^{0} - e^{0} + K))}^{\frac{1}{2}}$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify

${({(2 (0 e^{0} - e^{0} + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1

Multiply the exponents in

${({(2 (0 e^{0} - e^{0} + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3.2.1.1.2

Cancel the common factor of

$2$ .

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Step 5.3.2.1.1.2.1

Cancel the common factor.

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

Step 5.3.2.1.1.2.2

Rewrite the expression.

${(2 (0 e^{0} - e^{0} + K))}^{1} = 1^{2}$

Step 5.3.2.1.2

Simplify each term.

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Step 5.3.2.1.2.1

Anything raised to

$0$ is

$1$ .

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

Step 5.3.2.1.2.2

Multiply

$0$ by

$1$ .

${(2 (0 - e^{0} + K))}^{1} = 1^{2}$

Step 5.3.2.1.2.3

Anything raised to

$0$ is

$1$ .

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

Step 5.3.2.1.2.4

Multiply

$- 1$ by

$1$ .

${(2 (0 - 1 + K))}^{1} = 1^{2}$

Step 5.3.2.1.3

Simplify by multiplying through.

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Step 5.3.2.1.3.1

Subtract

$1$ from

$0$ .

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

Step 5.3.2.1.3.2

Apply the distributive property.

${(2 \cdot - 1 + 2 K)}^{1} = 1^{2}$

Step 5.3.2.1.3.3

Multiply.

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Step 5.3.2.1.3.3.1

Multiply

$2$ by

$- 1$ .

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

Step 5.3.2.1.3.3.2

Simplify.

$- 2 + 2 K = 1^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

One to any power is one.

$- 2 + 2 K = 1$

Step 5.4

Solve for

$K$ .

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Step 5.4.1

Move all terms not containing

$K$ to the right side of the equation.

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Step 5.4.1.1

Add

$2$ to both sides of the equation.

$2 K = 1 + 2$

Step 5.4.1.2

Add

$1$ and

$2$ .

$2 K = 3$

Step 5.4.2

Divide each term in

$2 K = 3$ by

$2$ and simplify.

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Step 5.4.2.1

Divide each term in

$2 K = 3$ by

$2$ .

$\frac{2 K}{2} = \frac{3}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of

$2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{3}{2}$

Step 5.4.2.2.1.2

Divide

$K$ by

$1$ .

$K = \frac{3}{2}$

Step 6

Substitute

$\frac{3}{2}$ for

$K$ in

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$ and simplify.

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Step 6.1

Substitute

$\frac{3}{2}$ for

$K$ .

$x = \sqrt{2 (t e^{t} - e^{t} + \frac{3}{2})}$

Step 6.2

Reorder terms.

$x = \sqrt{2 (e^{t} t + \frac{3}{2} - e^{t})}$

Step 6.3

To write

$e^{t} t$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$x = \sqrt{2 (e^{t} t \cdot \frac{2}{2} + \frac{3}{2} - e^{t})}$

Step 6.4

Combine

$e^{t} t$ and

$\frac{2}{2}$ .

$x = \sqrt{2 (\frac{e^{t} t \cdot 2}{2} + \frac{3}{2} - e^{t})}$

Step 6.5

Combine the numerators over the common denominator.

$x = \sqrt{2 (\frac{e^{t} t \cdot 2 + 3}{2} - e^{t})}$

Step 6.6

Move

$2$ to the left of

$e^{t} t$ .

$x = \sqrt{2 (\frac{2 e^{t} t + 3}{2} - e^{t})}$

Step 6.7

To write

$- e^{t}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$x = \sqrt{2 (\frac{2 e^{t} t + 3}{2} - e^{t} \cdot \frac{2}{2})}$

Step 6.8

Combine

$- e^{t}$ and

$\frac{2}{2}$ .

$x = \sqrt{2 (\frac{2 e^{t} t + 3}{2} + \frac{- e^{t} \cdot 2}{2})}$

Step 6.9

Combine the numerators over the common denominator.

$x = \sqrt{2 \frac{2 e^{t} t + 3 - e^{t} \cdot 2}{2}}$

Step 6.10

Multiply

$2$ by

$- 1$ .

$x = \sqrt{2 \frac{2 e^{t} t + 3 - 2 e^{t}}{2}}$

Step 6.11

Combine

$2$ and

$\frac{2 e^{t} t + 3 - 2 e^{t}}{2}$ .

$x = \sqrt{\frac{2 (2 e^{t} t + 3 - 2 e^{t})}{2}}$

Step 6.12

Reduce the expression by cancelling the common factors.

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Step 6.12.1

Reduce the expression

$\frac{2 (2 e^{t} t + 3 - 2 e^{t})}{2}$ by cancelling the common factors.

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Step 6.12.1.1

Cancel the common factor.

$x = \sqrt{\frac{2 (2 e^{t} t + 3 - 2 e^{t})}{2}}$

Step 6.12.1.2

Rewrite the expression.

$x = \sqrt{\frac{2 e^{t} t + 3 - 2 e^{t}}{1}}$

Step 6.12.2

Divide

$2 e^{t} t + 3 - 2 e^{t}$ by

$1$ .

$x = \sqrt{2 e^{t} t + 3 - 2 e^{t}}$

Step 6.13

Reorder factors in

$x = \sqrt{2 e^{t} t + 3 - 2 e^{t}}$ .

$x = \sqrt{2 t e^{t} + 3 - 2 e^{t}}$