Calculus Examples

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

Step 1

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

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 1.2

Factor out $\frac{y}{x}$ from $\frac{y (\ln (\frac{y}{x}) + 1)}{x}$ .

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Step 1.2.1

Factor $\frac{y}{x}$ out of $\frac{y (\ln (\frac{y}{x}) + 1)}{x}$ .

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

Step 1.2.2

Reorder $\frac{y}{x}$ and $\ln (\frac{y}{x}) + 1$ .

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

Step 2

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

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

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

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

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

Simplify $(\ln (V) + 1) V$ .

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Step 6.1.1.1.1

Rewrite.

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

Step 6.1.1.1.2

Simplify by adding zeros.

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

Step 6.1.1.1.3

Apply the distributive property.

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

Step 6.1.1.1.4

Simplify the expression.

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Step 6.1.1.1.4.1

Multiply $V$ by $1$ .

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

Step 6.1.1.1.4.2

Reorder factors in $\ln (V) V + V$ .

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

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.

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

Step 6.1.1.2.2

Combine the opposite terms in $V \ln (V) + V - V$ .

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Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

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

Step 6.1.1.2.2.2

Add $V \ln (V)$ and $0$ .

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

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = V \ln (V)$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = V \ln (V)$ by $x$ .

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

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.

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

Step 6.1.1.3.2.1.2

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

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

Step 6.1.2

Multiply both sides by $\frac{1}{V \ln (V)}$ .

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

Step 6.1.3

Cancel the common factor of $V \ln (V)$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.1.4

Rewrite the equation.

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

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Let $u = \ln (V)$ . Then $d u = \frac{1}{V} d V$ , so $V d u = d V$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.2.1.1

Let $u = \ln (V)$ . Find $\frac{d u}{d V}$ .

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Step 6.2.2.1.1.1

Differentiate $\ln (V)$ .

$\frac{d}{d V} [\ln (V)]$

Step 6.2.2.1.1.2

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

$\frac{1}{V}$

Step 6.2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.2

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

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

Step 6.2.2.3

Replace all occurrences of $u$ with $\ln (V)$ .

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.3

Solve for $V$ .

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Step 6.3.1

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

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

Step 6.3.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 6.3.3

To solve for $V$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| \ln (V) |}{| x |})} = e^{C}$

Step 6.3.4

Rewrite $\ln (\frac{| \ln (V) |}{| x |}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = \frac{| \ln (V) |}{| x |}$

Step 6.3.5

Solve for $V$ .

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Step 6.3.5.1

Rewrite the equation as $\frac{| \ln (V) |}{| x |} = e^{C}$ .

$\frac{| \ln (V) |}{| x |} = e^{C}$

Step 6.3.5.2

Multiply both sides by $| x |$ .

$\frac{| \ln (V) |}{| x |} | x | = e^{C} | x |$

Step 6.3.5.3

Simplify the left side.

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Step 6.3.5.3.1

Cancel the common factor of $| x |$ .

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Step 6.3.5.3.1.1

Cancel the common factor.

$\frac{| \ln (V) |}{| x |} | x | = e^{C} | x |$

Step 6.3.5.3.1.2

Rewrite the expression.

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

Step 6.3.5.4

Solve for $V$ .

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Step 6.3.5.4.1

Reorder factors in $e^{C} | x |$ .

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

Step 6.3.5.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$\ln (V) = \pm | x | e^{C}$

Step 6.3.5.4.3

To solve for $V$ , rewrite the equation using properties of logarithms.

$e^{\ln (V)} = e^{\pm | x | e^{C}}$

Step 6.3.5.4.4

Rewrite $\ln (V) = \pm | x | e^{C}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\pm | x | e^{C}} = V$

Step 6.3.5.4.5

Solve for $V$ .

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Step 6.3.5.4.5.1

Rewrite the equation as $V = e^{\pm | x | e^{C}}$ .

$V = e^{\pm | x | e^{C}}$

Step 6.3.5.4.5.2

Reorder factors in $e^{\pm | x | e^{C}}$ .

$V = e^{\pm e^{C} | x |}$

Step 6.4

Group the constant terms together.

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Step 6.4.1

Simplify the constant of integration.

$V = e^{\pm C | x |}$

Step 6.4.2

Combine constants with the plus or minus.

$V = e^{C | x |}$

Step 7

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

$\frac{y}{x} = e^{C | x |}$

Step 8

Solve $\frac{y}{x} = e^{C | x |}$ for $y$ .

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Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = e^{C | x |} x$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $x$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = e^{C | x |} x$

Step 8.2.1.1.2

Rewrite the expression.

$y = e^{C | x |} x$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Reorder factors in $e^{C | x |} x$ .

$y = x e^{C | x |}$