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

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

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Step 1.1.1

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

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

Step 1.1.2

Reorder $\frac{y}{x}$ and $\frac{y}{x + y}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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

Step 1.6

Simplify.

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

Step 1.7

Simplify.

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

Step 1.8

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

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

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x$ .

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Step 1.9.1.1

Cancel the common factor.

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

Step 1.9.1.2

Rewrite the expression.

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

Step 1.9.2

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

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

Step 2

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

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

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

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

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

Multiply $\frac{V}{1 + V} V$ .

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Step 6.1.1.1.1

Combine $\frac{V}{1 + V}$ and $V$ .

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

Step 6.1.1.1.2

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

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

Step 6.1.1.1.3

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

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

Step 6.1.1.1.4

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

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

Step 6.1.1.1.5

Add $1$ and $1$ .

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

Step 6.1.1.2

Subtract $V$ from both sides of the equation.

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

Step 6.1.1.3

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

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Step 6.1.1.3.1

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

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

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.

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

Step 6.1.1.3.2.1.2

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

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

Step 6.1.1.3.3

Simplify the right side.

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Step 6.1.1.3.3.1

Combine the numerators over the common denominator.

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

Step 6.1.1.3.3.2

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

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

Step 6.1.1.3.3.3

Simplify terms.

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Step 6.1.1.3.3.3.1

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

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

Step 6.1.1.3.3.3.2

Combine the numerators over the common denominator.

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

Step 6.1.1.3.3.4

Simplify the numerator.

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Step 6.1.1.3.3.4.1

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

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Step 6.1.1.3.3.4.1.1

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

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

Step 6.1.1.3.3.4.1.2

Factor $V$ out of $- V (1 + V)$ .

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

Step 6.1.1.3.3.4.1.3

Factor $V$ out of $V \cdot V + V (- (1 + V))$ .

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

Step 6.1.1.3.3.4.2

Apply the distributive property.

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

Step 6.1.1.3.3.4.3

Multiply $- 1$ by $1$ .

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

Step 6.1.1.3.3.4.4

Subtract $V$ from $V$ .

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

Step 6.1.1.3.3.4.5

Subtract $1$ from $0$ .

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

Step 6.1.1.3.3.5

Simplify the expression.

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Step 6.1.1.3.3.5.1

Move $- 1$ to the left of $V$ .

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

Step 6.1.1.3.3.5.2

Move the negative in front of the fraction.

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

Step 6.1.1.3.3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.1.1.3.3.7

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

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

Step 6.1.2

Regroup factors.

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

Step 6.1.3

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

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

Step 6.1.4

Simplify.

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Step 6.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.4.2

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

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

Step 6.1.4.3

Cancel the common factor of $1 + V$ .

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Step 6.1.4.3.1

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

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

Step 6.1.4.3.2

Factor $1 + V$ out of $- (1 + V)$ .

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

Step 6.1.4.3.3

Factor $1 + V$ out of $(1 + V) x$ .

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

Step 6.1.4.3.4

Cancel the common factor.

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

Step 6.1.4.3.5

Rewrite the expression.

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

Step 6.1.4.4

Cancel the common factor of $V$ .

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Step 6.1.4.4.1

Cancel the common factor.

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

Step 6.1.4.4.2

Rewrite the expression.

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

Step 6.1.5

Rewrite the equation.

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

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Split the fraction into multiple fractions.

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

Step 6.2.2.2

Split the single integral into multiple integrals.

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

Step 6.2.2.3

Cancel the common factor of $V$ .

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Step 6.2.2.3.1

Cancel the common factor.

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

Step 6.2.2.3.2

Rewrite the expression.

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

Step 6.2.2.4

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

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

Step 6.2.2.5

Apply the constant rule.

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

Step 6.2.2.6

Simplify.

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

Step 6.2.2.7

Reorder terms.

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

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.

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Simplify.

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

Step 6.2.4

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

$V + \ln (| V |) = - \ln (| x |) + C$

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

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.

$\ln (| \frac{y}{x} \cdot x |) = - \frac{y}{x} + C$

Step 8.3.2

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

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

Step 8.4

Cancel the common factor of $x$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Divide $y$ by $1$ .

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