Calculus Examples

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

Step 1

Rewrite the differential equation as a function of $\frac{y}{x}$ .

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Step 1.1

Split $\frac{y^{2} - x^{2}}{x y}$ and simplify.

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Step 1.1.1

Split the fraction $\frac{y^{2} - x^{2}}{x y}$ into two fractions.

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

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Cancel the common factors.

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Step 1.1.2.1.2.1

Factor $y$ out of $x y$ .

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

Step 1.1.2.1.2.2

Cancel the common factor.

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

Step 1.1.2.1.2.3

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.2.2.1

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

$\frac{d y}{d x} = \frac{y}{x} + \frac{x (- x)}{x y}$

Step 1.1.2.2.2

Cancel the common factors.

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Step 1.1.2.2.2.1

Factor $x$ out of $x y$ .

$\frac{d y}{d x} = \frac{y}{x} + \frac{x (- x)}{x (y)}$

Step 1.1.2.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{y}{x} + \frac{x (- x)}{x y}$

Step 1.1.2.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{y}{x} + \frac{- x}{y}$

Step 1.1.2.3

Move the negative in front of the fraction.

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

Step 1.2

Rewrite $\frac{x}{y}$ as ${(\frac{y}{x})}^{- 1}$ .

$\frac{d y}{d x} = \frac{y}{x} - {(\frac{y}{x})}^{- 1}$

Step 2

Let $V = \frac{y}{x}$ . Substitute $V$ for $\frac{y}{x}$ .

$\frac{d y}{d x} = V - V^{- 1}$

Step 3

Solve $V = \frac{y}{x}$ for $y$ .

$y = V x$

Step 4

Use the product rule to find the derivative of $y = V x$ with respect to $x$ .

$\frac{d y}{d x} = x \frac{d V}{d x} + V$

Step 5

Substitute $x \frac{d V}{d x} + V$ for $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = V - V^{- 1}$

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

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

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Step 6.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x \frac{d V}{d x} + V = V - \frac{1}{V}$

Step 6.1.1.2

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

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Step 6.1.1.2.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = V - \frac{1}{V} - V$

Step 6.1.1.2.2

Combine the opposite terms in $V - \frac{1}{V} - V$ .

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Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = - \frac{1}{V} + 0$

Step 6.1.1.2.2.2

Add $- \frac{1}{V}$ and $0$ .

$x \frac{d V}{d x} = - \frac{1}{V}$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = - \frac{1}{V}$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = - \frac{1}{V}$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{- \frac{1}{V}}{x}$

Step 6.1.1.3.2

Simplify the left side.

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Step 6.1.1.3.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{- \frac{1}{V}}{x}$

Step 6.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{- \frac{1}{V}}{x}$

Step 6.1.1.3.3

Simplify the right side.

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Step 6.1.1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = - \frac{1}{V} \cdot \frac{1}{x}$

Step 6.1.1.3.3.2

Multiply $\frac{1}{x}$ by $\frac{1}{V}$ .

$\frac{d V}{d x} = - \frac{1}{x V}$

Step 6.1.2

Regroup factors.

$\frac{d V}{d x} = - \frac{1}{x} \cdot \frac{1}{V}$

Step 6.1.3

Multiply both sides by $V$ .

$V \frac{d V}{d x} = V (- \frac{1}{x} \cdot \frac{1}{V})$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Rewrite using the commutative property of multiplication.

$V \frac{d V}{d x} = - V (\frac{1}{x} \cdot \frac{1}{V})$

Step 6.1.4.2

Multiply $\frac{1}{x}$ by $\frac{1}{V}$ .

$V \frac{d V}{d x} = - V \frac{1}{x V}$

Step 6.1.4.3

Cancel the common factor of $V$ .

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Step 6.1.4.3.1

Factor $V$ out of $- V$ .

$V \frac{d V}{d x} = V \cdot - 1 \frac{1}{x V}$

Step 6.1.4.3.2

Factor $V$ out of $x V$ .

$V \frac{d V}{d x} = V \cdot - 1 \frac{1}{V x}$

Step 6.1.4.3.3

Cancel the common factor.

$V \frac{d V}{d x} = V \cdot - 1 \frac{1}{V x}$

Step 6.1.4.3.4

Rewrite the expression.

$V \frac{d V}{d x} = - \frac{1}{x}$

Step 6.1.5

Rewrite the equation.

$V d V = - \frac{1}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int V d V = \int - \frac{1}{x} d x$

Step 6.2.2

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

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

Step 6.2.3

Integrate the right side.

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Step 6.2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6.2.3.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} V^{2} + C_{1} = - (\ln (| x |) + C_{2})$

Step 6.2.3.3

Simplify.

$\frac{1}{2} V^{2} + C_{1} = - \ln (| x |) + C_{2}$

Step 6.2.4

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

$\frac{1}{2} V^{2} = - \ln (| x |) + C$

Step 6.3

Solve for $V$ .

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Step 6.3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} V^{2}) = 2 (- \ln (| x |) + C)$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

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

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Step 6.3.2.1.1.1

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

$2 \frac{V^{2}}{2} = 2 (- \ln (| x |) + C)$

Step 6.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

$2 \frac{V^{2}}{2} = 2 (- \ln (| x |) + C)$

Step 6.3.2.1.1.2.2

Rewrite the expression.

$V^{2} = 2 (- \ln (| x |) + C)$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Simplify $2 (- \ln (| x |) + C)$ .

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Step 6.3.2.2.1.1

Apply the distributive property.

$V^{2} = 2 (- \ln (| x |)) + 2 C$

Step 6.3.2.2.1.2

Multiply $- 1$ by $2$ .

$V^{2} = - 2 \ln (| x |) + 2 C$

Step 6.3.3

Simplify $- 2 \ln (| x |)$ by moving $2$ inside the logarithm.

$V^{2} = - \ln ({| x |}^{2}) + 2 C$

Step 6.3.4

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

$V = \pm \sqrt{- \ln ({| x |}^{2}) + 2 C}$

Step 6.3.5

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$V = \pm \sqrt{- \ln (x^{2}) + 2 C}$

Step 6.3.6

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

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Step 6.3.6.1

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

$V = \sqrt{- \ln (x^{2}) + 2 C}$

Step 6.3.6.2

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

$V = - \sqrt{- \ln (x^{2}) + 2 C}$

Step 6.3.6.3

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

$V = \sqrt{- \ln (x^{2}) + 2 C}$

$V = - \sqrt{- \ln (x^{2}) + 2 C}$

$V = \sqrt{- \ln (x^{2}) + 2 C}$

$V = - \sqrt{- \ln (x^{2}) + 2 C}$

$V = \sqrt{- \ln (x^{2}) + 2 C}$

$V = - \sqrt{- \ln (x^{2}) + 2 C}$

Step 6.4

Simplify the constant of integration.

$V = \sqrt{- \ln (x^{2}) + C}$

$V = - \sqrt{- \ln (x^{2}) + C}$

$V = \sqrt{- \ln (x^{2}) + C}$

$V = - \sqrt{- \ln (x^{2}) + C}$

Step 7

Substitute $\frac{y}{x}$ for $V$ .

$\frac{y}{x} = \sqrt{- \ln (x^{2}) + C}, - \sqrt{- \ln (x^{2}) + C}$

Step 8

Solve $\frac{y}{x} = \sqrt{- \ln (x^{2}) + C}$ for $y$ .

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Step 8.1

Rewrite.

$\frac{y}{x} = \sqrt{- \ln (x^{2}) + C}$

Step 8.2

Multiply both sides by $x$ .

$\frac{y}{x} x = \sqrt{- \ln (x^{2}) + C} x$

Step 8.3

Simplify the left side.

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Step 8.3.1

Cancel the common factor of $x$ .

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Step 8.3.1.1

Cancel the common factor.

$\frac{y}{x} x = \sqrt{- \ln (x^{2}) + C} x$

Step 8.3.1.2

Rewrite the expression.

$y = \sqrt{- \ln (x^{2}) + C} x$

Step 9

Solve $\frac{y}{x} = - \sqrt{- \ln (x^{2}) + C}$ for $y$ .

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Step 9.1

Rewrite.

$\frac{y}{x} = - \sqrt{- \ln (x^{2}) + C}$

Step 9.2

Multiply both sides by $x$ .

$\frac{y}{x} x = - \sqrt{- \ln (x^{2}) + C} x$

Step 9.3

Simplify the left side.

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Step 9.3.1

Cancel the common factor of $x$ .

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Step 9.3.1.1

Cancel the common factor.

$\frac{y}{x} x = - \sqrt{- \ln (x^{2}) + C} x$

Step 9.3.1.2

Rewrite the expression.

$y = - \sqrt{- \ln (x^{2}) + C} x$

Step 10

List the solutions.

$y = \sqrt{- \ln (x^{2}) + C} x, y = - \sqrt{- \ln (x^{2}) + C} x$