Calculus Examples

$4 y \frac{d y}{d x} = 3 x^{2}$

Step 1

Rewrite the equation.

$4 y d y = 3 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 4 y d y = \int 3 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int y d y = \int 3 x^{2} d x$

Step 2.2.2

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

$4 (\frac{1}{2} y^{2} + C_{1}) = \int 3 x^{2} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $4 (\frac{1}{2} y^{2} + C_{1})$ as $4 (\frac{1}{2}) y^{2} + C_{1}$ .

$4 (\frac{1}{2}) y^{2} + C_{1} = \int 3 x^{2} d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} y^{2} + C_{1} = \int 3 x^{2} d x$

Step 2.2.3.2.2

Cancel the common factor of $4$ and $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $4$ .

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

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $3$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

Multiply $x^{3}$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{2} = x^{3} + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{2} = x^{3} + K$ by $2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Combine the numerators over the common denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Combine and simplify the denominator.

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Step 3.3.4.1

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

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

Step 3.3.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.4.4

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

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

Step 3.3.4.5

Add $1$ and $1$ .

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

Step 3.3.4.6

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

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Step 3.3.4.6.1

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

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

Step 3.3.4.6.2

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

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

Step 3.3.4.6.3

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

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

Step 3.3.4.6.4

Cancel the common factor of $2$ .

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Step 3.3.4.6.4.1

Cancel the common factor.

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

Step 3.3.4.6.4.2

Rewrite the expression.

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

Step 3.3.4.6.5

Evaluate the exponent.

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

Step 3.3.5

Combine using the product rule for radicals.

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

Step 3.3.6

Reorder factors in $\pm \frac{\sqrt{(x^{3} + K) \cdot 2}}{2}$ .

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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