Calculus Examples

$x e^{- t} \frac{d x}{d t} = t$ given the initial condition that $x (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x e^{- t} \frac{d x}{d t} = t$ by $x e^{- t}$ and simplify.

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Step 1.1.1

Divide each term in $x e^{- t} \frac{d x}{d t} = t$ by $x e^{- t}$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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

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