Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $e^{- t}$ is constant with respect to $x$ , move $e^{- t}$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $e^{- t}$ and $\frac{1}{2}$ .

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

Step 2.2.3.2.2

Combine $\frac{e^{- t}}{2}$ and $x^{2}$ .

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $\frac{1}{2} e^{- t} x^{2} = t y + K$ by $\frac{1}{2} e^{- t}$ and simplify.

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Step 3.1.1

Divide each term in $\frac{1}{2} e^{- t} x^{2} = t y + K$ by $\frac{1}{2} e^{- t}$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $\frac{1}{2}$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

Rewrite the expression.

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

Step 3.1.2.2

Cancel the common factor of $e^{- t}$ .

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Step 3.1.2.2.1

Cancel the common factor.

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

Step 3.1.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Combine $\frac{1}{2}$ and $e^{- t}$ .

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

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.3

Multiply $t y \frac{2}{e^{- t}}$ .

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Step 3.1.3.1.3.1

Combine $t$ and $\frac{2}{e^{- t}}$ .

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

Step 3.1.3.1.3.2

Combine $y$ and $\frac{t \cdot 2}{e^{- t}}$ .

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

Step 3.1.3.1.4

Move $2$ to the left of $y t$ .

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

Step 3.1.3.1.5

Combine $\frac{1}{2}$ and $e^{- t}$ .

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

Step 3.1.3.1.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.7

Combine $K$ and $\frac{2}{e^{- t}}$ .

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

Step 3.1.3.1.8

Move $2$ to the left of $K$ .

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

Step 3.2

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

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

Step 3.3

Simplify $\pm \sqrt{\frac{2 y t}{e^{- t}} + \frac{2 K}{e^{- t}}}$ .

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Step 3.3.1

Combine the numerators over the common denominator.

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

Step 3.3.2

Factor $2$ out of $2 y t + 2 K$ .

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Step 3.3.2.1

Factor $2$ out of $2 y t$ .

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

Step 3.3.2.2

Factor $2$ out of $2 K$ .

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

Step 3.3.2.3

Factor $2$ out of $2 (y t) + 2 K$ .

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

Step 3.3.3

Rewrite $\sqrt{\frac{2 (y t + K)}{e^{- t}}}$ as $\frac{\sqrt{2 (y t + K)}}{\sqrt{e^{- t}}}$ .

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

Step 3.3.4

Multiply $\frac{\sqrt{2 (y t + K)}}{\sqrt{e^{- t}}}$ by $\frac{\sqrt{e^{- t}}}{\sqrt{e^{- t}}}$ .

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

Step 3.3.5

Combine and simplify the denominator.

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Step 3.3.5.1

Multiply $\frac{\sqrt{2 (y t + K)}}{\sqrt{e^{- t}}}$ by $\frac{\sqrt{e^{- t}}}{\sqrt{e^{- t}}}$ .

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

Step 3.3.5.2

Raise $\sqrt{e^{- t}}$ to the power of $1$ .

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

Step 3.3.5.3

Raise $\sqrt{e^{- t}}$ to the power of $1$ .

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

Step 3.3.5.4

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

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

Step 3.3.5.5

Add $1$ and $1$ .

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

Step 3.3.5.6

Rewrite ${\sqrt{e^{- t}}}^{2}$ as $e^{- t}$ .

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Step 3.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e^{- t}}$ as $e^{\frac{- t}{2}}$ .

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

Step 3.3.5.6.2

Move the negative in front of the fraction.

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

Step 3.3.5.6.3

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

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

Step 3.3.5.6.4

Multiply $2$ by $- 1$ .

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

Step 3.3.5.6.5

Combine $- 2$ and $\frac{t}{2}$ .

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

Step 3.3.5.6.6

Cancel the common factor of $- 2$ and $2$ .

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Step 3.3.5.6.6.1

Factor $2$ out of $- 2 t$ .

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

Step 3.3.5.6.6.2

Cancel the common factors.

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Step 3.3.5.6.6.2.1

Factor $2$ out of $2$ .

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

Step 3.3.5.6.6.2.2

Cancel the common factor.

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

Step 3.3.5.6.6.2.3

Rewrite the expression.

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

Step 3.3.5.6.6.2.4

Divide $- t$ by $1$ .

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

Step 3.3.6

Combine using the product rule for radicals.

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

Step 3.3.7

Reorder factors in $\pm \frac{\sqrt{2 (y t + K) e^{- t}}}{e^{- t}}$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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