Calculus Examples

$\frac{d f}{d t} = y (f - c)$

Step 1

Let $v = f - c$ . Substitute $v$ for all occurrences of $f - c$ .

$\frac{d f}{d t} = y v$

Step 2

Find $\frac{d v}{d t}$ by differentiating $f - c$ .

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $f - c$ with respect to $t$ is $\frac{d}{d t} [f] + \frac{d}{d t} [- c]$ .

$\frac{d v}{d t} = \frac{d}{d t} [f] + \frac{d}{d t} [- c]$

Step 2.2

Rewrite $\frac{d}{d t} [f]$ as $f'$ .

$\frac{d v}{d t} = f' + \frac{d}{d t} [- c]$

Step 2.3

Since $- c$ is constant with respect to $t$ , the derivative of $- c$ with respect to $t$ is $0$ .

$\frac{d v}{d t} = f' + 0$

Step 2.4

Add $f'$ and $0$ .

$\frac{d v}{d t} = f'$

Step 3

Substitute the derivative back in to the differential equation.

$\frac{d v}{d t} = y v$

Step 4

Separate the variables.

Tap for more steps...

Step 4.1

Multiply both sides by $\frac{1}{v}$ .

$\frac{1}{v} \frac{d v}{d t} = \frac{1}{v} (y v)$

Step 4.2

Cancel the common factor of $v$ .

Tap for more steps...

Step 4.2.1

Factor $v$ out of $y v$ .

$\frac{1}{v} \frac{d v}{d t} = \frac{1}{v} (v y)$

Step 4.2.2

Cancel the common factor.

$\frac{1}{v} \frac{d v}{d t} = \frac{1}{v} (v y)$

Step 4.2.3

Rewrite the expression.

$\frac{1}{v} \frac{d v}{d t} = y$

Step 4.3

Rewrite the equation.

$\frac{1}{v} d v = y d t$

Step 5

Integrate both sides.

Tap for more steps...

Step 5.1

Set up an integral on each side.

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

Step 5.2

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

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

Step 5.3

Integrate the right side.

Tap for more steps...

Step 5.3.1

Apply the constant rule.

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

Step 5.3.2

Reorder terms.

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

Step 5.4

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

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

Step 6

Solve for $v$ .

Tap for more steps...

Step 6.1

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

$e^{\ln (| v |)} = e^{t y + C}$

Step 6.2

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

$e^{t y + C} = | v |$

Step 6.3

Solve for $v$ .

Tap for more steps...

Step 6.3.1

Rewrite the equation as $| v | = e^{t y + C}$ .

$| v | = e^{t y + C}$

Step 6.3.2

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

$v = \pm e^{t y + C}$

Step 7

Group the constant terms together.

Tap for more steps...

Step 7.1

Rewrite $e^{t y + C}$ as $e^{t y} e^{C}$ .

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

Step 7.2

Reorder $e^{t y}$ and $e^{C}$ .

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

Step 7.3

Combine constants with the plus or minus.

$v = C e^{t y}$

Step 8

Replace all occurrences of $v$ with $f - c$ .

$f - c = C e^{t y}$

Step 9

Add $c$ to both sides of the equation.

$f = C e^{t y} + c$