Calculus Examples

$\frac{1}{y} \cdot \frac{d y}{d t} = 2 \cdot \frac{1}{t}$

Step 1

Rewrite the equation.

$\frac{1}{y} \cdot d y = 2 \cdot \frac{1}{t} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 2 \frac{1}{t} d t$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 2.3.3

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

$\ln (| y |) + C_{1} = 2 (\ln (| t |) + C_{2})$

Step 2.3.4

Simplify.

$\ln (| y |) + C_{1} = 2 \ln (| t |) + C_{2}$

Step 2.4

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

$\ln (| y |) = 2 \ln (| t |) + C$

Step 3

Solve for $y$ .

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Step 3.1

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

$\ln (| y |) - 2 \ln (| t |) = C$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $\ln (| y |) - 2 \ln (| t |)$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

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

$\ln (| y |) - \ln ({| t |}^{2}) = C$

Step 3.2.1.1.2

Remove the absolute value in ${| t |}^{2}$ because exponentiations with even powers are always positive.

$\ln (| y |) - \ln (t^{2}) = C$

Step 3.2.1.2

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

$\ln (\frac{| y |}{t^{2}}) = C$

Step 3.3

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

$e^{\ln (\frac{| y |}{t^{2}})} = e^{C}$

Step 3.4

Rewrite $\ln (\frac{| y |}{t^{2}}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{| y |}{t^{2}}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $\frac{| y |}{t^{2}} = e^{C}$ .

$\frac{| y |}{t^{2}} = e^{C}$

Step 3.5.2

Multiply both sides by $t^{2}$ .

$\frac{| y |}{t^{2}} t^{2} = e^{C} t^{2}$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Cancel the common factor of $t^{2}$ .

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Step 3.5.3.1.1.1

Cancel the common factor.

$\frac{| y |}{t^{2}} t^{2} = e^{C} t^{2}$

Step 3.5.3.1.1.2

Rewrite the expression.

$| y | = e^{C} t^{2}$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Reorder factors in $e^{C} t^{2}$ .

$| y | = t^{2} e^{C}$

Step 3.5.4

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

$y = \pm t^{2} e^{C}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \pm t^{2} C$

Step 4.2

Combine constants with the plus or minus.

$y = C t^{2}$