Calculus Examples

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

Step 1

Rewrite the differential equation.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Rewrite the expression.

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

Step 2.1.2.2

Cancel the common factor of $x$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Cancel the common factor of $y$ .

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Step 2.1.2.3.1

Cancel the common factor.

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

Step 2.1.2.3.2

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

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

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

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Step 2.1.3.1.1.1

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

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

Step 2.1.3.1.1.2

Cancel the common factors.

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Step 2.1.3.1.1.2.1

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

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

Step 2.1.3.1.1.2.2

Cancel the common factor.

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

Step 2.1.3.1.1.2.3

Rewrite the expression.

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

Step 2.1.3.1.2

Cancel the common factor of $2$ .

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Step 2.1.3.1.2.1

Cancel the common factor.

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

Step 2.1.3.1.2.2

Rewrite the expression.

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

Step 2.1.3.1.3

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

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Step 2.1.3.1.3.1

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

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

Step 2.1.3.1.3.2

Cancel the common factors.

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Step 2.1.3.1.3.2.1

Factor $y$ out of $x y$ .

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

Step 2.1.3.1.3.2.2

Cancel the common factor.

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

Step 2.1.3.1.3.2.3

Rewrite the expression.

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

Step 2.2

Rewrite the differential equation as $f (\frac{y}{x})$ .

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Step 2.2.1

Factor out $\frac{x}{y}$ from $\frac{x}{2 y}$ .

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Step 2.2.1.1

Factor $\frac{x}{y}$ out of $\frac{x}{2 y}$ .

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

Step 2.2.1.2

Reorder $\frac{x}{y}$ and $\frac{1}{2}$ .

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

Step 2.2.2

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

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

Step 3

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

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

Step 4

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

$y = V x$

Step 5

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 6

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

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

Step 7

Solve the substituted differential equation.

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Step 7.1

Separate the variables.

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Step 7.1.1

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

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Step 7.1.1.1

Simplify each term.

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Step 7.1.1.1.1

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

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

Step 7.1.1.1.2

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

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

Step 7.1.1.2

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

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Step 7.1.1.2.1

Subtract $V$ from both sides of the equation.

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

Step 7.1.1.2.2

Combine the opposite terms in $\frac{1}{2 V} + V - V$ .

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Step 7.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \frac{1}{2 V} + 0$

Step 7.1.1.2.2.2

Add $\frac{1}{2 V}$ and $0$ .

$x \frac{d V}{d x} = \frac{1}{2 V}$

Step 7.1.1.3

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

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Step 7.1.1.3.1

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

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{2 V}}{x}$

Step 7.1.1.3.2

Simplify the left side.

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Step 7.1.1.3.2.1

Cancel the common factor of $x$ .

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Step 7.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{1}{2 V}}{x}$

Step 7.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{\frac{1}{2 V}}{x}$

Step 7.1.1.3.3

Simplify the right side.

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Step 7.1.1.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = \frac{1}{2 V} \cdot \frac{1}{x}$

Step 7.1.1.3.3.2

Multiply $\frac{1}{2 V}$ by $\frac{1}{x}$ .

$\frac{d V}{d x} = \frac{1}{2 V x}$

Step 7.1.2

Regroup factors.

$\frac{d V}{d x} = \frac{1}{2 V} \cdot \frac{1}{x}$

Step 7.1.3

Multiply both sides by $2 V$ .

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

Step 7.1.4

Simplify.

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Step 7.1.4.1

Multiply $\frac{1}{2 V}$ by $\frac{1}{x}$ .

$2 V \frac{d V}{d x} = 2 V \frac{1}{2 V x}$

Step 7.1.4.2

Cancel the common factor of $2 V$ .

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Step 7.1.4.2.1

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

$2 V \frac{d V}{d x} = 2 V \frac{1}{2 V (x)}$

Step 7.1.4.2.2

Cancel the common factor.

$2 V \frac{d V}{d x} = 2 V \frac{1}{2 V x}$

Step 7.1.4.2.3

Rewrite the expression.

$2 V \frac{d V}{d x} = \frac{1}{x}$

Step 7.1.5

Rewrite the equation.

$2 V d V = \frac{1}{x} d x$

Step 7.2

Integrate both sides.

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Step 7.2.1

Set up an integral on each side.

$\int 2 V d V = \int \frac{1}{x} d x$

Step 7.2.2

Integrate the left side.

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Step 7.2.2.1

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

$2 \int V d V = \int \frac{1}{x} d x$

Step 7.2.2.2

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

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

Step 7.2.2.3

Simplify the answer.

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Step 7.2.2.3.1

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

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

Step 7.2.2.3.2

Simplify.

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Step 7.2.2.3.2.1

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

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

Step 7.2.2.3.2.2

Cancel the common factor of $2$ .

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Step 7.2.2.3.2.2.1

Cancel the common factor.

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

Step 7.2.2.3.2.2.2

Rewrite the expression.

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

Step 7.2.2.3.2.3

Multiply $V^{2}$ by $1$ .

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

Step 7.2.3

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

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

Step 7.2.4

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

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

Step 7.3

Solve for $V$ .

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Step 7.3.1

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

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

Step 7.3.2

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

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Step 7.3.2.1

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

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

Step 7.3.2.2

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