Calculus Examples

$x y \frac{d y}{d x} + 3 = 0$ , $f (2) = 1$

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Subtract $3$ from both sides of the equation.

$x y \frac{d y}{d x} = - 3$

Step 1.1.2

Divide each term in $x y \frac{d y}{d x} = - 3$ by $x y$ and simplify.

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Step 1.1.2.1

Divide each term in $x y \frac{d y}{d x} = - 3$ by $x y$ .

$\frac{x y \frac{d y}{d x}}{x y} = \frac{- 3}{x y}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{- 3}{x y}$

Step 1.1.2.2.1.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{- 3}{x y}$

Step 1.1.2.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{- 3}{x y}$

Step 1.1.2.2.2.2

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

$\frac{d y}{d x} = \frac{- 3}{x y}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{3}{x y}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = - \frac{3}{x} \cdot \frac{1}{y}$

Step 1.3

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (- \frac{3}{x} \cdot \frac{1}{y})$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = - y (\frac{3}{x} \cdot \frac{1}{y})$

Step 1.4.2

Multiply $\frac{3}{x}$ by $\frac{1}{y}$ .

$y \frac{d y}{d x} = - y \frac{3}{x y}$

Step 1.4.3

Cancel the common factor of $y$ .

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Step 1.4.3.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{3}{x y}$

Step 1.4.3.2

Factor $y$ out of $x y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{3}{y x}$

Step 1.4.3.3

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{3}{y x}$

Step 1.4.3.4

Rewrite the expression.

$y \frac{d y}{d x} = - \frac{3}{x}$

Step 1.5

Rewrite the equation.

$y d y = - \frac{3}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - \frac{3}{x} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.3

Multiply $3$ by $- 1$ .

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.4

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

$\frac{1}{2} y^{2} = - 3 \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- 3 \ln (| x |) + C)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (- 3 \ln (| x |) + C)$

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.

$2 \frac{y^{2}}{2} = 2 (- 3 \ln (| x |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- 3 \ln (| x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- 3 \ln (| x |) + C)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{2} = 2 (- 3 \ln (| x |)) + 2 C$

Step 3.2.2.1.2

Multiply $- 3$ by $2$ .

$y^{2} = - 6 \ln (| x |) + 2 C$

Step 3.3

Simplify $- 6 \ln (| x |)$ by moving $6$ inside the logarithm.

$y^{2} = - \ln ({| x |}^{6}) + 2 C$

Step 3.4

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

$y = \pm \sqrt{- \ln ({| x |}^{6}) + 2 C}$

Step 3.5

Remove the absolute value in ${| x |}^{6}$ because exponentiations with even powers are always positive.

$y = \pm \sqrt{- \ln (x^{6}) + 2 C}$

Step 3.6

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

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Step 3.6.1

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

$y = \sqrt{- \ln (x^{6}) + 2 C}$

Step 3.6.2

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

$y = - \sqrt{- \ln (x^{6}) + 2 C}$

Step 3.6.3

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

$y = \sqrt{- \ln (x^{6}) + 2 C}$

$y = - \sqrt{- \ln (x^{6}) + 2 C}$

$y = \sqrt{- \ln (x^{6}) + 2 C}$

$y = - \sqrt{- \ln (x^{6}) + 2 C}$

$y = \sqrt{- \ln (x^{6}) + 2 C}$

$y = - \sqrt{- \ln (x^{6}) + 2 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt{- \ln (x^{6}) + C}$

$y = - \sqrt{- \ln (x^{6}) + C}$

Step 5

Since $y$ is positive in the initial condition $(2, 1)$ , only consider $y = \sqrt{- \ln (x^{6}) + C}$ to find the $C$ . Substitute $2$ for $x$ and $1$ for $y$ .

$1 = \sqrt{- \ln (2^{6}) + C}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $\sqrt{- \ln (2^{6}) + C} = 1$ .

$\sqrt{- \ln (2^{6}) + C} = 1$

Step 6.2

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

${\sqrt{- \ln (2^{6}) + C}}^{2} = 1^{2}$

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 $\sqrt{- \ln (2^{6}) + C}$ as ${(- \ln (2^{6}) + C)}^{\frac{1}{2}}$ .

${({(- \ln (2^{6}) + C)}^{\frac{1}{2}})}^{2} = 1^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${({(- \ln (2^{6}) + C)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1

Multiply the exponents in ${({(- \ln (2^{6}) + C)}^{\frac{1}{2}})}^{2}$ .

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Step 6.3.2.1.1.1

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

${(- \ln (2^{6}) + C)}^{\frac{1}{2} \cdot 2} = 1^{2}$

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.

${(- \ln (2^{6}) + C)}^{\frac{1}{2} \cdot 2} = 1^{2}$

Step 6.3.2.1.1.2.2

Rewrite the expression.

${(- \ln (2^{6}) + C)}^{1} = 1^{2}$

Step 6.3.2.1.2

Raise $2$ to the power of $6$ .

${(- \ln (64) + C)}^{1} = 1^{2}$

Step 6.3.2.1.3

Simplify.

$- \ln (64) + C = 1^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

One to any power is one.

$- \ln (64) + C = 1$

Step 6.4

Add $\ln (64)$ to both sides of the equation.

$C = 1 + \ln (64)$

Step 7

Substitute $1 + \ln (64)$ for $C$ in $y = \sqrt{- \ln (x^{6}) + C}$ and simplify.

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Step 7.1

Substitute $1 + \ln (64)$ for $C$ .

$y = \sqrt{- \ln (x^{6}) + 1 + \ln (64)}$