Calculus Examples

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

Step 1

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

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

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

Step 2.2.3.2.2.2

Rewrite the expression.

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

Step 2.2.3.2.3

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

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Simplify $\pm \sqrt{- \frac{1}{3} x^{3} + K}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Simplify terms.

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Step 3.2.3.1

Combine $K$ and $\frac{3}{3}$ .

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

Step 3.2.3.2

Combine the numerators over the common denominator.

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

Step 3.2.4

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

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

Step 3.2.5

Rewrite $\sqrt{\frac{- x^{3} + 3 K}{3}}$ as $\frac{\sqrt{- x^{3} + 3 K}}{\sqrt{3}}$ .

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

Step 3.2.6

Multiply $\frac{\sqrt{- x^{3} + 3 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.7

Combine and simplify the denominator.

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Step 3.2.7.1

Multiply $\frac{\sqrt{- x^{3} + 3 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.7.2

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

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

Step 3.2.7.3

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

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

Step 3.2.7.4

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

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

Step 3.2.7.5

Add $1$ and $1$ .

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

Step 3.2.7.6

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

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Step 3.2.7.6.1

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

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

Step 3.2.7.6.2

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

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

Step 3.2.7.6.3

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

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

Step 3.2.7.6.4

Cancel the common factor of $2$ .

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Step 3.2.7.6.4.1

Cancel the common factor.

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

Step 3.2.7.6.4.2

Rewrite the expression.

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

Step 3.2.7.6.5

Evaluate the exponent.

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

Step 3.2.8

Combine using the product rule for radicals.

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

Step 3.2.9

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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