Calculus Examples

$\frac{d y}{d x} = \frac{\ln (x)}{x y}$ , $y (1) = 2$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $\frac{\ln (x)}{x}$ by $\frac{1}{y}$ .

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Factor $y$ out of $x y$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $\ln (x)$ .

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

Step 2.3.1.1.2

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

$\frac{1}{x}$

Step 2.3.1.2

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

$\frac{1}{2} y^{2} + C_{1} = \int u d u$

Step 2.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} u^{2} + C_{2}$

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{2} \ln^{2} (x) + K)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{2}$ and $\ln^{2} (x)$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.2

Rewrite the expression.

$y^{2} = \ln^{2} (x) + 2 K$

Step 3.3

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

$y = \pm \sqrt{\ln^{2} (x) + 2 K}$

Step 3.4

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

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Step 3.4.1

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

$y = \sqrt{\ln^{2} (x) + 2 K}$

Step 3.4.2

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

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

Step 3.4.3

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

$y = \sqrt{\ln^{2} (x) + 2 K}$

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

$y = \sqrt{\ln^{2} (x) + 2 K}$

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

$y = \sqrt{\ln^{2} (x) + 2 K}$

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

Step 4

Simplify the constant of integration.

$y = \sqrt{\ln^{2} (x) + K}$

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

Step 5

Since $y$ is positive in the initial condition $(1, 2)$ , only consider $y = \sqrt{\ln^{2} (x) + K}$ to find the $K$ . Substitute $1$ for $x$ and $2$ for $y$ .

$2 = \sqrt{\ln^{2} (1) + K}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\sqrt{\ln^{2} (1) + K} = 2$ .

$\sqrt{\ln^{2} (1) + K} = 2$

Step 6.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\ln^{2} (1) + K}}^{2} = 2^{2}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\ln^{2} (1) + K}$ as ${(\ln^{2} (1) + K)}^{\frac{1}{2}}$ .

${({(\ln^{2} (1) + K)}^{\frac{1}{2}})}^{2} = 2^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${({(\ln^{2} (1) + K)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1

Multiply the exponents in ${({(\ln^{2} (1) + K)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1.1

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

${(\ln^{2} (1) + K)}^{\frac{1}{2} \cdot 2} = 2^{2}$

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.

${(\ln^{2} (1) + K)}^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 6.3.2.1.1.2.2

Rewrite the expression.

${(\ln^{2} (1) + K)}^{1} = 2^{2}$

Step 6.3.2.1.2

Simplify each term.

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Step 6.3.2.1.2.1

The natural logarithm of $1$ is $0$ .

${(0^{2} + K)}^{1} = 2^{2}$

Step 6.3.2.1.2.2

Raising $0$ to any positive power yields $0$ .

${(0 + K)}^{1} = 2^{2}$

Step 6.3.2.1.3

Simplify by adding zeros.

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Step 6.3.2.1.3.1

Add $0$ and $K$ .

$K^{1} = 2^{2}$

Step 6.3.2.1.3.2

Simplify.

$K = 2^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Raise $2$ to the power of $2$ .

$K = 4$

Step 7

Substitute $4$ for $K$ in $y = \sqrt{\ln^{2} (x) + K}$ and simplify.

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Step 7.1

Substitute $4$ for $K$ .

$y = \sqrt{\ln^{2} (x) + 4}$