Calculus Examples

$x^{2} d x + y e d y = 0$

Step 1

Subtract $x^{2} d x$ from both sides of the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $e$ is constant with respect to $y$ , move $e$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Combine $e$ and $\frac{1}{2}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{e}{2} y^{2} + C_{1} = - (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Rewrite $- (\frac{1}{3} x^{3} + C_{2})$ as $- \frac{1}{3} x^{3} + C_{2}$ .

$\frac{e}{2} y^{2} + C_{1} = - \frac{1}{3} x^{3} + C_{2}$

Step 2.4

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

$\frac{e}{2} y^{2} = - \frac{1}{3} x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{2}{e}$ .

$\frac{2}{e} (\frac{e}{2} y^{2}) = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

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

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Step 3.2.1.1.1

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

$\frac{2}{e} \cdot \frac{e y^{2}}{2} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.1.1.2

Combine.

$\frac{2 (e y^{2})}{e \cdot 2} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$\frac{2 (e y^{2})}{e \cdot 2} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.1.1.3.2

Rewrite the expression.

$\frac{e y^{2}}{e} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.1.1.4

Cancel the common factor of $e$ .

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Step 3.2.1.1.4.1

Cancel the common factor.

$\frac{e y^{2}}{e} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.1.1.4.2

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

$y^{2} = \frac{2}{e} (- \frac{1}{3} x^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{2}{e} (- \frac{1}{3} x^{3} + K)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Combine $x^{3}$ and $\frac{1}{3}$ .

$y^{2} = \frac{2}{e} (- \frac{x^{3}}{3} + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{2} = \frac{2}{e} (- \frac{x^{3}}{3}) + \frac{2}{e} K$

Step 3.2.2.1.1.3

Multiply $\frac{2}{e}$ by $\frac{x^{3}}{3}$ .

$y^{2} = - \frac{2 x^{3}}{e \cdot 3} + \frac{2}{e} K$

Step 3.2.2.1.1.4

Combine $\frac{2}{e}$ and $K$ .

$y^{2} = - \frac{2 x^{3}}{e \cdot 3} + \frac{2 K}{e}$

Step 3.2.2.1.2

Move $3$ to the left of $e$ .

$y^{2} = - \frac{2 x^{3}}{3 e} + \frac{2 K}{e}$

Step 3.3

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

$y = \pm \sqrt{- \frac{2 x^{3}}{3 e} + \frac{2 K}{e}}$

Step 3.4

Simplify $\pm \sqrt{- \frac{2 x^{3}}{3 e} + \frac{2 K}{e}}$ .

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Step 3.4.1

To write $\frac{2 K}{e}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{- \frac{2 x^{3}}{3 e} + \frac{2 K}{e} \cdot \frac{3}{3}}$

Step 3.4.2

Write each expression with a common denominator of $3 e$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

Multiply $\frac{2 K}{e}$ by $\frac{3}{3}$ .

$y = \pm \sqrt{- \frac{2 x^{3}}{3 e} + \frac{2 K \cdot 3}{e \cdot 3}}$

Step 3.4.2.2

Reorder the factors of $e \cdot 3$ .

$y = \pm \sqrt{- \frac{2 x^{3}}{3 e} + \frac{2 K \cdot 3}{3 e}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- 2 x^{3} + 2 K \cdot 3}{3 e}}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Factor $2$ out of $- 2 x^{3} + 2 K \cdot 3$ .

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Step 3.4.4.1.1

Factor $2$ out of $- 2 x^{3}$ .

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

Step 3.4.4.1.2

Factor $2$ out of $2 K \cdot 3$ .

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

Step 3.4.4.1.3

Factor $2$ out of $2 (- x^{3}) + 2 (K \cdot 3)$ .

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

Step 3.4.4.2

Move $3$ to the left of $K$ .

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

Step 3.4.5

Rewrite $\sqrt{\frac{2 (- x^{3} + 3 K)}{3 e}}$ as $\frac{\sqrt{2 (- x^{3} + 3 K)}}{\sqrt{3 e}}$ .

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

Step 3.4.6

Multiply $\frac{\sqrt{2 (- x^{3} + 3 K)}}{\sqrt{3 e}}$ by $\frac{\sqrt{3 e}}{\sqrt{3 e}}$ .

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

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{\sqrt{2 (- x^{3} + 3 K)}}{\sqrt{3 e}}$ by $\frac{\sqrt{3 e}}{\sqrt{3 e}}$ .

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

Step 3.4.7.2

Raise $\sqrt{3 e}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{\sqrt{3 e}}^{1} \sqrt{3 e}}$

Step 3.4.7.3

Raise $\sqrt{3 e}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{\sqrt{3 e}}^{1} {\sqrt{3 e}}^{1}}$

Step 3.4.7.4

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

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{\sqrt{3 e}}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

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

Step 3.4.7.6

Rewrite ${\sqrt{3 e}}^{2}$ as $3 e$ .

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Step 3.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 e}$ as ${(3 e)}^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{({(3 e)}^{\frac{1}{2}})}^{2}}$

Step 3.4.7.6.2

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

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{(3 e)}^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.6.3

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

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

Step 3.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.4.7.6.4.1

Cancel the common factor.

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

Step 3.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 (- x^{3} + 3 K)} \sqrt{3 e}}{{(3 e)}^{1}}$

Step 3.4.7.6.5

Simplify.

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

Step 3.4.8

Simplify the numerator.

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Step 3.4.8.1

Combine using the product rule for radicals.

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

Step 3.4.8.2

Multiply $3$ by $2$ .

$y = \pm \frac{\sqrt{6 (- x^{3} + 3 K) e}}{3 e}$

Step 3.4.9

Reorder factors in $\pm \frac{\sqrt{6 (- x^{3} + 3 K) e}}{3 e}$ .

$y = \pm \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

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.

$y = \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

Step 3.5.2

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

$y = - \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

Step 3.5.3

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

$y = \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

$y = - \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

$y = \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

$y = - \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

$y = \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

$y = - \frac{\sqrt{6 e (- x^{3} + 3 K)}}{3 e}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 e (- x^{3} + K)}}{3 e}$

$y = - \frac{\sqrt{6 e (- x^{3} + K)}}{3 e}$