Calculus Examples

$\frac{d y}{d x} = - 2 y$ , $y (2) = 3$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (- 2 y)$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y} \frac{d y}{d x} = - 2 \frac{1}{y} y$

Step 1.2.2

Combine $- 2$ and $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{- 2}{y} y$

Step 1.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.3.1

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{- 2}{y} y$

Step 1.2.3.2

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = - 2$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = - 2 d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int - 2 d x$

Step 2.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

$\ln (| y |) = - 2 x + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$e^{- 2 x + C} = | y |$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $| y | = e^{- 2 x + C}$ .

$| y | = e^{- 2 x + C}$

Step 3.3.2

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

$y = \pm e^{- 2 x + C}$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Rewrite $e^{- 2 x + C}$ as $e^{- 2 x} e^{C}$ .

$y = \pm e^{- 2 x} e^{C}$

Step 4.2

Reorder $e^{- 2 x}$ and $e^{C}$ .

$y = \pm e^{C} e^{- 2 x}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{- 2 x}$

Step 5

Use the initial condition to find the value of $C$ by substituting $2$ for $x$ and $3$ for $y$ in $y = C e^{- 2 x}$ .

$3 = C e^{- 2 \cdot 2}$

Step 6

Solve for $C$ .

Tap for more steps...

Step 6.1

Rewrite the equation as $C e^{- 2 \cdot 2} = 3$ .

$C e^{- 2 \cdot 2} = 3$

Step 6.2

Multiply $- 2$ by $2$ .

$C e^{- 4} = 3$

Step 6.3

Divide each term in $C e^{- 4} = 3$ by $e^{- 4}$ and simplify.

Tap for more steps...

Step 6.3.1

Divide each term in $C e^{- 4} = 3$ by $e^{- 4}$ .

$\frac{C e^{- 4}}{e^{- 4}} = \frac{3}{e^{- 4}}$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $e^{- 4}$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

$\frac{C e^{- 4}}{e^{- 4}} = \frac{3}{e^{- 4}}$

Step 6.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{3}{e^{- 4}}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Move $e^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$C = 3 e^{4}$

Step 7

Substitute $3 e^{4}$ for $C$ in $y = C e^{- 2 x}$ and simplify.

Tap for more steps...

Step 7.1

Substitute $3 e^{4}$ for $C$ .

$y = 3 e^{4} e^{- 2 x}$

Step 7.2

Multiply $e^{4}$ by $e^{- 2 x}$ by adding the exponents.

Tap for more steps...

Step 7.2.1

Move $e^{- 2 x}$ .

$y = 3 (e^{- 2 x} e^{4})$

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 3 e^{- 2 x + 4}$