Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.2

Multiply $\frac{x y}{x^{2} + y^{2}}$ by $\frac{\frac{1}{x^{2}}}{\frac{1}{x^{2}}}$ .

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

Step 1.3

Multiply $\frac{x y}{x^{2} + y^{2}}$ by $\frac{\frac{1}{x^{2}}}{\frac{1}{x^{2}}}$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Cancel the common factor of $x$ .

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Step 1.7.1

Factor $x$ out of $x y$ .

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

Step 1.7.2

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

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

Step 1.7.3

Cancel the common factor.

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

Step 1.7.4

Rewrite the expression.

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

Step 1.8

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

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

Step 1.9

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

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

Step 2

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

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

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

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

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

Subtract $V$ from both sides of the equation.

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

Step 6.1.1.2

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

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Step 6.1.1.2.1

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

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

Step 6.1.1.2.2

Simplify the left side.

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Step 6.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.2.2.1.1

Cancel the common factor.

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

Step 6.1.1.2.2.1.2

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

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

Step 6.1.1.2.3

Simplify the right side.

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Step 6.1.1.2.3.1

Combine the numerators over the common denominator.

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

Step 6.1.1.2.3.2

To write $- V$ as a fraction with a common denominator, multiply by $\frac{1 + V^{2}}{1 + V^{2}}$ .

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

Step 6.1.1.2.3.3

Simplify terms.

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Step 6.1.1.2.3.3.1

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

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

Step 6.1.1.2.3.3.2

Combine the numerators over the common denominator.

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

Step 6.1.1.2.3.4

Simplify the numerator.

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Step 6.1.1.2.3.4.1

Factor $V$ out of $V - V (1 + V^{2})$ .

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Step 6.1.1.2.3.4.1.1

Raise $V$ to the power of $1$ .

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

Step 6.1.1.2.3.4.1.2

Factor $V$ out of $V^{1}$ .

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

Step 6.1.1.2.3.4.1.3

Factor $V$ out of $- V (1 + V^{2})$ .

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

Step 6.1.1.2.3.4.1.4

Factor $V$ out of $V \cdot 1 + V (- (1 + V^{2}))$ .

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

Step 6.1.1.2.3.4.2

Apply the distributive property.

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

Step 6.1.1.2.3.4.3

Multiply $- 1$ by $1$ .

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

Step 6.1.1.2.3.4.4

Subtract $1$ from $1$ .

$\frac{d V}{d x} = \frac{\frac{V (0 - V^{2})}{1 + V^{2}}}{x}$

Step 6.1.1.2.3.4.5

Subtract $V^{2}$ from $0$ .

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

Step 6.1.1.2.3.4.6

Combine exponents.

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Step 6.1.1.2.3.4.6.1

Factor out negative.

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

Step 6.1.1.2.3.4.6.2

Multiply $V$ by $V^{2}$ by adding the exponents.

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Step 6.1.1.2.3.4.6.2.1

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

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Step 6.1.1.2.3.4.6.2.1.1

Raise $V$ to the power of $1$ .

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

Step 6.1.1.2.3.4.6.2.1.2

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

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

Step 6.1.1.2.3.4.6.2.2

Add $1$ and $2$ .

$\frac{d V}{d x} = \frac{\frac{- V^{3}}{1 + V^{2}}}{x}$

Step 6.1.1.2.3.5

Move the negative in front of the fraction.

$\frac{d V}{d x} = \frac{- \frac{V^{3}}{1 + V^{2}}}{x}$

Step 6.1.1.2.3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{d V}{d x} = - \frac{V^{3}}{1 + V^{2}} \cdot \frac{1}{x}$

Step 6.1.1.2.3.7

Multiply $\frac{1}{x}$ by $\frac{V^{3}}{1 + V^{2}}$ .

$\frac{d V}{d x} = - \frac{V^{3}}{x (1 + V^{2})}$

Step 6.1.2

Regroup factors.

$\frac{d V}{d x} = - \frac{V^{3}}{1 + V^{2}} \cdot \frac{1}{x}$

Step 6.1.3

Multiply both sides by $\frac{1 + V^{2}}{V^{3}}$ .

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{1 + V^{2}}{V^{3}} (- \frac{V^{3}}{1 + V^{2}} \cdot \frac{1}{x})$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Rewrite using the commutative property of multiplication.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = - \frac{1 + V^{2}}{V^{3}} (\frac{V^{3}}{1 + V^{2}} \cdot \frac{1}{x})$

Step 6.1.4.2

Multiply $\frac{V^{3}}{1 + V^{2}}$ by $\frac{1}{x}$ .

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = - \frac{1 + V^{2}}{V^{3}} \cdot \frac{V^{3}}{(1 + V^{2}) x}$

Step 6.1.4.3

Cancel the common factor of $1 + V^{2}$ .

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Step 6.1.4.3.1

Move the leading negative in $- \frac{1 + V^{2}}{V^{3}}$ into the numerator.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{- (1 + V^{2})}{V^{3}} \cdot \frac{V^{3}}{(1 + V^{2}) x}$

Step 6.1.4.3.2

Factor $1 + V^{2}$ out of $- (1 + V^{2})$ .

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{(1 + V^{2}) \cdot - 1}{V^{3}} \cdot \frac{V^{3}}{(1 + V^{2}) x}$

Step 6.1.4.3.3

Factor $1 + V^{2}$ out of $(1 + V^{2}) x$ .

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{(1 + V^{2}) \cdot - 1}{V^{3}} \cdot \frac{V^{3}}{(1 + V^{2}) (x)}$

Step 6.1.4.3.4

Cancel the common factor.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{(1 + V^{2}) \cdot - 1}{V^{3}} \cdot \frac{V^{3}}{(1 + V^{2}) x}$

Step 6.1.4.3.5

Rewrite the expression.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{- 1}{V^{3}} \cdot \frac{V^{3}}{x}$

Step 6.1.4.4

Cancel the common factor of $V^{3}$ .

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Step 6.1.4.4.1

Cancel the common factor.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = \frac{- 1}{V^{3}} \cdot \frac{V^{3}}{x}$

Step 6.1.4.4.2

Rewrite the expression.

$\frac{1 + V^{2}}{V^{3}} \frac{d V}{d x} = - \frac{1}{x}$

Step 6.1.5

Rewrite the equation.

$\frac{1 + V^{2}}{V^{3}} d V = - \frac{1}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1 + V^{2}}{V^{3}} d V = \int - \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Apply basic rules of exponents.

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Step 6.2.2.1.1

Move $V^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2.2.1.2

Multiply the exponents in ${(V^{3})}^{- 1}$ .

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Step 6.2.2.1.2.1

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

$\int (1 + V^{2}) V^{3 \cdot - 1} d V = \int - \frac{1}{x} d x$

Step 6.2.2.1.2.2

Multiply $3$ by $- 1$ .

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

Step 6.2.2.2

Multiply $(1 + V^{2}) V^{- 3}$ .

$\int 1 V^{- 3} + V^{2} V^{- 3} d V = \int - \frac{1}{x} d x$

Step 6.2.2.3

Simplify.

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Step 6.2.2.3.1

Multiply $V^{- 3}$ by $1$ .

$\int V^{- 3} + V^{2} V^{- 3} d V = \int - \frac{1}{x} d x$

Step 6.2.2.3.2

Multiply $V^{2}$ by $V^{- 3}$ by adding the exponents.

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Step 6.2.2.3.2.1

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

$\int V^{- 3} + V^{2 - 3} d V = \int - \frac{1}{x} d x$

Step 6.2.2.3.2.2

Subtract $3$ from $2$ .

$\int V^{- 3} + V^{- 1} d V = \int - \frac{1}{x} d x$

Step 6.2.2.4

Split the single integral into multiple integrals.

$\int V^{- 3} d V + \int V^{- 1} d V = \int - \frac{1}{x} d x$

Step 6.2.2.5

By the Power Rule, the integral of $V^{- 3}$ with respect to $V$ is $- \frac{1}{2} V^{- 2}$ .

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

Step 6.2.2.6

The integral of $V^{- 1}$ with respect to $V$ is $\ln (| V |)$ .

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

Step 6.2.2.7

Simplify.

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Step 6.2.2.7.1

Simplify.

$- \frac{1}{2} \cdot \frac{1}{V^{2}} + \ln (| V |) + C_{3} = \int - \frac{1}{x} d x$

Step 6.2.2.7.2

Simplify.

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Step 6.2.2.7.2.1

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

$- \frac{1}{V^{2} \cdot 2} + \ln (| V |) + C_{3} = \int - \frac{1}{x} d x$

Step 6.2.2.7.2.2

Move $2$ to the left of $V^{2}$ .

$- \frac{1}{2 V^{2}} + \ln (| V |) + C_{3} = \int - \frac{1}{x} d x$

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.

$- \frac{1}{2 V^{2}} + \ln (| V |) + C_{3} = - \int \frac{1}{x} d x$

Step 6.2.3.2

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

$- \frac{1}{2 V^{2}} + \ln (| V |) + C_{3} = - (\ln (| x |) + C_{4})$

Step 6.2.3.3

Simplify.

$- \frac{1}{2 V^{2}} + \ln (| V |) + C_{3} = - \ln (| x |) + C_{4}$

Step 6.2.4

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

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

Step 7

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

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

Step 8

Solve $- \frac{1}{2 {(\frac{y}{x})}^{2}} + \ln (| \frac{y}{x} |) = - \ln (| x |) + C$ for $y$ .

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Step 8.1

Move all the terms containing a logarithm to the left side of the equation.

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

Step 8.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 8.3

Multiply $| \frac{y}{x} | | x |$ .

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Step 8.3.1

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 8.3.2

Combine $\frac{y}{x}$ and $x$ .

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

Step 8.4

Cancel the common factor of $x$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Divide $y$ by $1$ .

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

Step 8.5

Simplify each term.

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Step 8.5.1

Apply the product rule to $\frac{y}{x}$ .

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

Step 8.5.2

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

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

Step 8.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.5.4

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

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