Calculus Examples

$\frac{d y}{d x} = \frac{x^{2}}{y}$ , $y (0) = 3$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{x^{2}}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{2}}{y}$

Step 1.2.2

Rewrite the expression.

$y \frac{d y}{d x} = x^{2}$

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Combine $2$ and $\frac{x^{3}}{3}$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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Step 3.4.1.1

Factor $2$ out of $\frac{2 x^{3}}{3}$ .

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

Step 3.4.1.2

Factor $2$ out of $2 K$ .

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

Step 3.4.1.3

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

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

Step 3.4.2

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

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

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

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

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

Step 3.4.3.2

Combine the numerators over the common denominator.

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

Step 3.4.4

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

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

Step 3.4.5

Combine $2$ and $\frac{x^{3} + 3 K}{3}$ .

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

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

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

Step 3.4.8.2

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

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

Step 3.4.8.3

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

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

Step 3.4.8.4

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

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

Step 3.4.8.5

Add $1$ and $1$ .

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

Step 3.4.8.6

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

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Step 3.4.8.6.1

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

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

Step 3.4.8.6.2

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

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

Step 3.4.8.6.3

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

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

Step 3.4.8.6.4

Cancel the common factor of $2$ .

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Step 3.4.8.6.4.1

Cancel the common factor.

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

Step 3.4.8.6.4.2

Rewrite the expression.

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

Step 3.4.8.6.5

Evaluate the exponent.

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

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Combine using the product rule for radicals.

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

Step 3.4.9.2

Multiply $3$ by $2$ .

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

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.

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

Step 3.5.2

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

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

Step 3.5.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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

Step 5

Since $y$ is positive in the initial condition $(0, 3)$ , only consider $y = \frac{\sqrt{6 (x^{3} + K)}}{3}$ to find the $K$ . Substitute $0$ for $x$ and $3$ for $y$ .

$3 = \frac{\sqrt{6 (0^{3} + K)}}{3}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\frac{\sqrt{6 (0^{3} + K)}}{3} = 3$ .

$\frac{\sqrt{6 (0^{3} + K)}}{3} = 3$

Step 6.2

Multiply both sides by $3$ .

$\frac{\sqrt{6 (0^{3} + K)}}{3} \cdot 3 = 3 \cdot 3$

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{\sqrt{6 (0^{3} + K)}}{3} \cdot 3$ .

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Step 6.3.1.1.1

Simplify the numerator.

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Step 6.3.1.1.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{\sqrt{6 (0 + K)}}{3} \cdot 3 = 3 \cdot 3$

Step 6.3.1.1.1.2

Add $0$ and $K$ .

$\frac{\sqrt{6 K}}{3} \cdot 3 = 3 \cdot 3$

Step 6.3.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.1.1.2.1

Cancel the common factor.

$\frac{\sqrt{6 K}}{3} \cdot 3 = 3 \cdot 3$

Step 6.3.1.1.2.2

Rewrite the expression.

$\sqrt{6 K} = 3 \cdot 3$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $3$ by $3$ .

$\sqrt{6 K} = 9$

Step 6.4

Solve for $K$ .

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Step 6.4.1

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

${\sqrt{6 K}}^{2} = 9^{2}$

Step 6.4.2

Simplify each side of the equation.

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Step 6.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 K}$ as ${(6 K)}^{\frac{1}{2}}$ .

${({(6 K)}^{\frac{1}{2}})}^{2} = 9^{2}$

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Simplify ${({(6 K)}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.2.2.1.1

Multiply the exponents in ${({(6 K)}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.2.2.1.1.1

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

${(6 K)}^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 6.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.2.2.1.1.2.1

Cancel the common factor.

${(6 K)}^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 6.4.2.2.1.1.2.2

Rewrite the expression.

${(6 K)}^{1} = 9^{2}$

Step 6.4.2.2.1.2

Simplify.

$6 K = 9^{2}$

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Raise $9$ to the power of $2$ .

$6 K = 81$

Step 6.4.3

Divide each term in $6 K = 81$ by $6$ and simplify.

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Step 6.4.3.1

Divide each term in $6 K = 81$ by $6$ .

$\frac{6 K}{6} = \frac{81}{6}$

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $6$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

$\frac{6 K}{6} = \frac{81}{6}$

Step 6.4.3.2.1.2

Divide $K$ by $1$ .

$K = \frac{81}{6}$

Step 6.4.3.3

Simplify the right side.

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Step 6.4.3.3.1

Cancel the common factor of $81$ and $6$ .

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Step 6.4.3.3.1.1

Factor $3$ out of $81$ .

$K = \frac{3 (27)}{6}$

Step 6.4.3.3.1.2

Cancel the common factors.

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Step 6.4.3.3.1.2.1

Factor $3$ out of $6$ .

$K = \frac{3 \cdot 27}{3 \cdot 2}$

Step 6.4.3.3.1.2.2

Cancel the common factor.

$K = \frac{3 \cdot 27}{3 \cdot 2}$

Step 6.4.3.3.1.2.3

Rewrite the expression.

$K = \frac{27}{2}$

Step 7

Substitute $\frac{27}{2}$ for $K$ in $y = \frac{\sqrt{6 (x^{3} + K)}}{3}$ and simplify.

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Step 7.1

Substitute $\frac{27}{2}$ for $K$ .

$y = \frac{\sqrt{6 (x^{3} + \frac{27}{2})}}{3}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

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

$y = \frac{\sqrt{6 (x^{3} \cdot \frac{2}{2} + \frac{27}{2})}}{3}$

Step 7.2.2

Combine $x^{3}$ and $\frac{2}{2}$ .

$y = \frac{\sqrt{6 (\frac{x^{3} \cdot 2}{2} + \frac{27}{2})}}{3}$

Step 7.2.3

Combine the numerators over the common denominator.

$y = \frac{\sqrt{6 \frac{x^{3} \cdot 2 + 27}{2}}}{3}$

Step 7.2.4

Move $2$ to the left of $x^{3}$ .

$y = \frac{\sqrt{6 \frac{2 x^{3} + 27}{2}}}{3}$

Step 7.2.5

Combine $6$ and $\frac{2 x^{3} + 27}{2}$ .

$y = \frac{\sqrt{\frac{6 (2 x^{3} + 27)}{2}}}{3}$

Step 7.2.6

Reduce the expression by cancelling the common factors.

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Step 7.2.6.1

Reduce the expression $\frac{6 (2 x^{3} + 27)}{2}$ by cancelling the common factors.

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Step 7.2.6.1.1

Factor $2$ out of $6 (2 x^{3} + 27)$ .

$y = \frac{\sqrt{\frac{2 (3 (2 x^{3} + 27))}{2}}}{3}$

Step 7.2.6.1.2

Factor $2$ out of $2$ .

$y = \frac{\sqrt{\frac{2 (3 (2 x^{3} + 27))}{2 (1)}}}{3}$

Step 7.2.6.1.3

Cancel the common factor.

$y = \frac{\sqrt{\frac{2 (3 (2 x^{3} + 27))}{2 \cdot 1}}}{3}$

Step 7.2.6.1.4

Rewrite the expression.

$y = \frac{\sqrt{\frac{3 (2 x^{3} + 27)}{1}}}{3}$

Step 7.2.6.2

Divide $3 (2 x^{3} + 27)$ by $1$ .

$y = \frac{\sqrt{3 (2 x^{3} + 27)}}{3}$