Calculus Examples

$\frac{1}{y} \cdot \frac{d y}{d t} = 5 \cdot \frac{1}{t}$

Step 1

Rewrite the equation.

$\frac{1}{y} \cdot d y = 5 \cdot \frac{1}{t} d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 5 \frac{1}{t} d t$

Step 2.2

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

$\ln (| y |) + C_{1} = \int 5 \frac{1}{t} d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$\ln (| y |) + C_{1} = \int \frac{5}{t} d t$

Step 2.3.2

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$\ln (| y |) + C_{1} = 5 \int \frac{1}{t} d t$

Step 2.3.3

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

$\ln (| y |) + C_{1} = 5 (\ln (| t |) + C_{2})$

Step 2.3.4

Simplify.

$\ln (| y |) + C_{1} = 5 \ln (| t |) + C_{2}$

Step 2.4

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

$\ln (| y |) = 5 \ln (| t |) + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

$\ln (| y |) - 5 \ln (| t |) = C$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify $\ln (| y |) - 5 \ln (| t |)$ .

Tap for more steps...

Step 3.2.1.1

Simplify $- 5 \ln (| t |)$ by moving $5$ inside the logarithm.

$\ln (| y |) - \ln ({| t |}^{5}) = C$

Step 3.2.1.2

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

$\ln (\frac{| y |}{{| t |}^{5}}) = C$

Step 3.3

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

$e^{\ln (\frac{| y |}{{| t |}^{5}})} = e^{C}$

Step 3.4

Rewrite $\ln (\frac{| y |}{{| t |}^{5}}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{| y |}{{| t |}^{5}}$

Step 3.5

Solve for $y$ .

Tap for more steps...

Step 3.5.1

Rewrite the equation as $\frac{| y |}{{| t |}^{5}} = e^{C}$ .

$\frac{| y |}{{| t |}^{5}} = e^{C}$

Step 3.5.2

Multiply both sides by ${| t |}^{5}$ .

$\frac{| y |}{{| t |}^{5}} {| t |}^{5} = e^{C} {| t |}^{5}$

Step 3.5.3

Simplify the left side.

Tap for more steps...

Step 3.5.3.1

Cancel the common factor of ${| t |}^{5}$ .

Tap for more steps...

Step 3.5.3.1.1

Cancel the common factor.

$\frac{| y |}{{| t |}^{5}} {| t |}^{5} = e^{C} {| t |}^{5}$

Step 3.5.3.1.2

Rewrite the expression.

$| y | = e^{C} {| t |}^{5}$

Step 3.5.4

Solve for $y$ .

Tap for more steps...

Step 3.5.4.1

Reorder factors in $e^{C} {| t |}^{5}$ .

$| y | = {| t |}^{5} e^{C}$

Step 3.5.4.2

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

$y = \pm {| t |}^{5} e^{C}$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Simplify the constant of integration.

$y = \pm {| t |}^{5} C$

Step 4.2

Combine constants with the plus or minus.

$y = C {| t |}^{5}$