Calculus Examples

$2 \frac{d y}{d x} - \frac{1}{y} = \frac{2 x}{y}$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Subtract $\frac{2 x}{y}$ from both sides of the equation.

$2 \frac{d y}{d x} - \frac{1}{y} - \frac{2 x}{y} = 0$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Add $\frac{1}{y}$ to both sides of the equation.

$2 \frac{d y}{d x} - \frac{2 x}{y} = \frac{1}{y}$

Step 1.1.2.2

Add $\frac{2 x}{y}$ to both sides of the equation.

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

Step 1.1.3

Divide each term in $2 \frac{d y}{d x} = \frac{1}{y} + \frac{2 x}{y}$ by $2$ and simplify.

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Step 1.1.3.1

Divide each term in $2 \frac{d y}{d x} = \frac{1}{y} + \frac{2 x}{y}$ by $2$ .

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $2$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.3.1.2

Multiply $\frac{1}{y}$ by $\frac{1}{2}$ .

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

Step 1.1.3.3.1.3

Move $2$ to the left of $y$ .

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

Step 1.1.3.3.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.3.1.5

Cancel the common factor of $2$ .

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Step 1.1.3.3.1.5.1

Factor $2$ out of $2 x$ .

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

Step 1.1.3.3.1.5.2

Cancel the common factor.

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

Step 1.1.3.3.1.5.3

Rewrite the expression.

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

Step 1.2

Factor.

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Step 1.2.1

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

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

Step 1.2.2

Write each expression with a common denominator of $2 y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

Multiply $\frac{x}{y}$ by $\frac{2}{2}$ .

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

Step 1.2.2.2

Reorder the factors of $y \cdot 2$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Move $2$ to the left of $x$ .

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

Step 1.3

Multiply both sides by $y$ .

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

Step 1.4

Cancel the common factor of $y$ .

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Step 1.4.1

Factor $y$ out of $2 y$ .

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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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.

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

Step 3.2.2.1.1.2

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

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

Step 3.2.2.1.1.3

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

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

Step 3.2.2.1.3.1.2

Rewrite the expression.

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

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

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

Step 3.2.2.1.3.2.2

Rewrite the expression.

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

Step 3.3

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

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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