Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $3 y$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

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

Step 2.3.6.2

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

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

Step 3.2.2.1.1.2

Combine $\frac{1}{3}$ and $x^{2}$ .

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

Step 3.2.2.1.1.3

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

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

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

Step 3.2.2.1.3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Combine the numerators over the common denominator.

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

Step 3.4.2

Factor $2 x$ out of $2 x^{2} + 2 x$ .

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Step 3.4.2.1

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

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

Step 3.4.2.2

Factor $2 x$ out of $2 x$ .

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

Step 3.4.2.3

Factor $2 x$ out of $2 x (x) + 2 x (1)$ .

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

Step 3.4.3

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

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

Step 3.4.4

Simplify terms.

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Step 3.4.4.1

Combine $2 K$ and $\frac{3}{3}$ .

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

Step 3.4.4.2

Combine the numerators over the common denominator.

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Factor $2$ out of $2 x (x + 1) + 2 K \cdot 3$ .

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Step 3.4.5.1.1

Factor $2$ out of $2 x (x + 1)$ .

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

Step 3.4.5.1.2

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

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

Step 3.4.5.1.3

Factor $2$ out of $2 (x (x + 1)) + 2 (K \cdot 3)$ .

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

Step 3.4.5.2

Apply the distributive property.

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

Step 3.4.5.3

Multiply $x$ by $x$ .

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

Step 3.4.5.4

Multiply $x$ by $1$ .

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

Step 3.4.5.5

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

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

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

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

Step 3.4.8.2

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

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

Step 3.4.8.3

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

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

Step 3.4.8.4

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

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

Step 3.4.8.5

Add $1$ and $1$ .

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

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 $3^{\frac{1}{2}}$ .

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

Step 3.4.8.6.2

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

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

Step 3.4.8.6.3

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

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

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.

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

Step 3.4.8.6.4.2

Rewrite the expression.

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

Step 3.4.8.6.5

Evaluate the exponent.

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

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Combine using the product rule for radicals.

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

Step 3.4.9.2

Multiply $3$ by $2$ .

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

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.

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

Step 3.5.2

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

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

Step 3.5.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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