Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

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

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

Step 1.1.2.2

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

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

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

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

Step 6.1.2

Regroup factors.

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

Step 6.1.3

Multiply both sides by $V$ .

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

Step 6.1.4

Simplify.

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Step 6.1.4.1

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

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

Step 6.1.4.2

Cancel the common factor of $V$ .

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Step 6.1.4.2.1

Factor $V$ out of $V x$ .

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

Step 6.1.4.2.2

Cancel the common factor.

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

Step 6.1.4.2.3

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

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 6.2.4

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

Apply the distributive property.

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

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.

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