Calculus Examples

$\frac{d y}{d x} = - \frac{1}{5 x y}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = - \frac{1}{5 x} \cdot \frac{1}{y}$

Step 1.2

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (- \frac{1}{5 x} \cdot \frac{1}{y})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = - y (\frac{1}{5 x} \cdot \frac{1}{y})$

Step 1.3.2

Multiply $\frac{1}{5 x}$ by $\frac{1}{y}$ .

$y \frac{d y}{d x} = - y \frac{1}{5 x y}$

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{1}{5 x y}$

Step 1.3.3.2

Factor $y$ out of $5 x y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{1}{y (5 x)}$

Step 1.3.3.3

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{1}{y (5 x)}$

Step 1.3.3.4

Rewrite the expression.

$y \frac{d y}{d x} = - \frac{1}{5 x}$

Step 1.4

Rewrite the equation.

$y d y = - \frac{1}{5 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - \frac{1}{5 x} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int - \frac{1}{5 x} d x$

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.

$\frac{1}{2} y^{2} + C_{1} = - \int \frac{1}{5 x} d x$

Step 2.3.2

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - (\frac{1}{5} \int \frac{1}{x} d x)$

Step 2.3.3

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

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{5} (\ln (| x |) + C_{2})$

Step 2.3.4

Simplify.

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{5} \ln (| x |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{1}{5} \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- \frac{1}{5} \ln (| x |) + C)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{5} \ln (| x |) + C)$

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.

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{5} \ln (| x |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- \frac{1}{5} \ln (| x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{1}{5} \ln (| x |) + C)$ .

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Step 3.2.2.1.1

Combine $\ln (| x |)$ and $\frac{1}{5}$ .

$y^{2} = 2 (- \frac{\ln (| x |)}{5} + C)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 (- \frac{\ln (| x |)}{5}) + 2 C$

Step 3.2.2.1.3

Multiply $2 (- \frac{\ln (| x |)}{5})$ .

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Step 3.2.2.1.3.1

Multiply $- 1$ by $2$ .

$y^{2} = - 2 \frac{\ln (| x |)}{5} + 2 C$

Step 3.2.2.1.3.2

Combine $- 2$ and $\frac{\ln (| x |)}{5}$ .

$y^{2} = \frac{- 2 \ln (| x |)}{5} + 2 C$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$y^{2} = - \frac{2 \ln (| x |)}{5} + 2 C$

Step 3.3

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$y^{2} = - \frac{\ln ({| x |}^{2})}{5} + 2 C$

Step 3.4

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

$y = \pm \sqrt{- \frac{\ln ({| x |}^{2})}{5} + 2 C}$

Step 3.5

Simplify $\pm \sqrt{- \frac{\ln ({| x |}^{2})}{5} + 2 C}$ .

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Step 3.5.1

Rewrite $\frac{\ln ({| x |}^{2})}{5}$ as $\frac{1}{5} \ln ({| x |}^{2})$ .

$y = \pm \sqrt{- (\frac{1}{5} \ln ({| x |}^{2})) + 2 C}$

Step 3.5.2

Simplify $\frac{1}{5} \ln ({| x |}^{2})$ by moving $\frac{1}{5}$ inside the logarithm.

$y = \pm \sqrt{- \ln ({({| x |}^{2})}^{\frac{1}{5}}) + 2 C}$

Step 3.5.3

Multiply the exponents in ${({| x |}^{2})}^{\frac{1}{5}}$ .

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Step 3.5.3.1

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

$y = \pm \sqrt{- \ln ({| x |}^{2 (\frac{1}{5})}) + 2 C}$

Step 3.5.3.2

Combine $2$ and $\frac{1}{5}$ .

$y = \pm \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

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.

$y = \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

Step 3.6.2

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

$y = - \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

Step 3.6.3

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

$y = \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

$y = - \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

$y = \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

$y = - \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

$y = \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

$y = - \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + 2 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + C}$

$y = - \sqrt{- \ln ({| x |}^{\frac{2}{5}}) + C}$