Calculus Examples

$\frac{d A}{d r} = A b^{2} \cos (b r)$ , $A (0) = b^{3}$

Step 1

Rewrite the differential equation as $\frac{d A}{d r} + M (r) A = Q (r)$ .

Tap for more steps...

Step 1.1

Rewrite the equation as $M (r) \frac{d A}{d r} + P (r) A = Q (r)$ .

Tap for more steps...

Step 1.1.1

Subtract $A b^{2} \cos (b r)$ from both sides of the equation.

$\frac{d A}{d r} - A b^{2} \cos (b r) = 0$

Step 1.1.2

Reorder terms.

$\frac{d A}{d r} - b^{2} A \cos (b r) = 0$

Step 1.2

Factor $A$ out of $- b^{2} A \cos (b r)$ .

$\frac{d A}{d r} + A (- b^{2} \cos (b r)) = 0$

Step 1.3

Reorder $A$ and $- b^{2} \cos (b r)$ .

$\frac{d A}{d r} - b^{2} \cos (b r) A = 0$

Step 2

The integrating factor is defined by the formula $e^{\int P (r) d r}$ , where $P (r) = - b^{2} \cos (b r)$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int - b^{2} \cos (b r) d r}$

Step 2.2

Integrate $- b^{2} \cos (b r)$ .

Tap for more steps...

Step 2.2.1

Since $- b^{2}$ is constant with respect to $r$ , move $- b^{2}$ out of the integral.

$e^{- b^{2} \int \cos (b r) d r}$

Step 2.2.2

Let $u = b r$ . Then $d u = b d r$ , so $\frac{1}{b} d u = d r$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.2.1

Let $u = b r$ . Find $\frac{d u}{d r}$ .

Tap for more steps...

Step 2.2.2.1.1

Rewrite.

$1 b$

Step 2.2.2.1.2

Multiply $b$ by $1$ .

$b$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$e^{- b^{2} \int \cos (u) \frac{1}{b} d u}$

Step 2.2.3

Combine $\cos (u)$ and $\frac{1}{b}$ .

$e^{- b^{2} \int \frac{\cos (u)}{b} d u}$

Step 2.2.4

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

$e^{- b^{2} (\frac{1}{b} \int \cos (u) d u)}$

Step 2.2.5

Simplify.

Tap for more steps...

Step 2.2.5.1

Combine $\frac{1}{b}$ and $b^{2}$ .

$e^{- \frac{b^{2}}{b} \int \cos (u) d u}$

Step 2.2.5.2

Cancel the common factor of $b^{2}$ and $b$ .

Tap for more steps...

Step 2.2.5.2.1

Factor $b$ out of $b^{2}$ .

$e^{- \frac{b \cdot b}{b} \int \cos (u) d u}$

Step 2.2.5.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.5.2.2.1

Raise $b$ to the power of $1$ .

$e^{- \frac{b \cdot b}{b^{1}} \int \cos (u) d u}$

Step 2.2.5.2.2.2

Factor $b$ out of $b^{1}$ .

$e^{- \frac{b \cdot b}{b \cdot 1} \int \cos (u) d u}$

Step 2.2.5.2.2.3

Cancel the common factor.

$e^{- \frac{b \cdot b}{b \cdot 1} \int \cos (u) d u}$

Step 2.2.5.2.2.4

Rewrite the expression.

$e^{- \frac{b}{1} \int \cos (u) d u}$

Step 2.2.5.2.2.5

Divide $b$ by $1$ .

$e^{- b \int \cos (u) d u}$

Step 2.2.6

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$e^{- b (\sin (u) + C)}$

Step 2.2.7

Simplify.

$e^{- b \sin (u) + C}$

Step 2.2.8

Replace all occurrences of $u$ with $b r$ .

$e^{- b \sin (b r) + C}$

Step 2.3

Remove the constant of integration.

$e^{- b \sin (b r)}$

Step 3

Multiply each term by the integrating factor $e^{- b \sin (b r)}$ .

Tap for more steps...

Step 3.1

Multiply each term by $e^{- b \sin (b r)}$ .

$e^{- b \sin (b r)} \frac{d A}{d r} + e^{- b \sin (b r)} (- b^{2} \cos (b r) A) = e^{- b \sin (b r)} \cdot 0$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- b \sin (b r)} \frac{d A}{d r} - e^{- b \sin (b r)} b^{2} \cos (b r) A = e^{- b \sin (b r)} \cdot 0$

Step 3.3

Multiply $e^{- b \sin (b r)}$ by $0$ .

$e^{- b \sin (b r)} \frac{d A}{d r} - e^{- b \sin (b r)} b^{2} \cos (b r) A = 0$

Step 3.4

Reorder factors in $e^{- b \sin (b r)} \frac{d A}{d r} - e^{- b \sin (b r)} b^{2} \cos (b r) A = 0$ .

$e^{- b \sin (b r)} \frac{d A}{d r} - b^{2} A e^{- b \sin (b r)} \cos (b r) = 0$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d r} [e^{- b \sin (b r)} A] = 0$

Step 5

Set up an integral on each side.

$\int \frac{d}{d r} [e^{- b \sin (b r)} A] d r = \int 0 d r$

Step 6

Integrate the left side.

$e^{- b \sin (b r)} A = \int 0 d r$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

The integral of $0$ with respect to $r$ is $0$ .

$e^{- b \sin (b r)} A = 0 + C$

Step 7.2

Add $0$ and $C$ .

$e^{- b \sin (b r)} A = C$

Step 8

Divide each term in $e^{- b \sin (b r)} A = C$ by $e^{- b \sin (b r)}$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $e^{- b \sin (b r)} A = C$ by $e^{- b \sin (b r)}$ .

$\frac{e^{- b \sin (b r)} A}{e^{- b \sin (b r)}} = \frac{C}{e^{- b \sin (b r)}}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $e^{- b \sin (b r)}$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{e^{- b \sin (b r)} A}{e^{- b \sin (b r)}} = \frac{C}{e^{- b \sin (b r)}}$

Step 8.2.1.2

Divide $A$ by $1$ .

$A = \frac{C}{e^{- b \sin (b r)}}$

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $r$ and $b^{3}$ for $A$ in $A = \frac{C}{e^{- b \sin (b r)}}$ .

$b^{3} = \frac{C}{e^{- b \sin (b \cdot 0)}}$

Step 10

Solve for $b$ .

Tap for more steps...

Step 10.1

Multiply both sides by $e^{- b \sin (b \cdot 0)}$ .

$b^{3} e^{- b \sin (b \cdot 0)} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

Simplify the left side.

Tap for more steps...

Step 10.2.1.1

Simplify $b^{3} e^{- b \sin (b \cdot 0)}$ .

Tap for more steps...

Step 10.2.1.1.1

Multiply $b$ by $0$ .

$b^{3} e^{- b \sin (0)} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.1.1.2

The exact value of $\sin (0)$ is $0$ .

$b^{3} e^{- b \cdot 0} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.1.1.3

Multiply $- b \cdot 0$ .

Tap for more steps...

Step 10.2.1.1.3.1

Multiply $0$ by $- 1$ .

$b^{3} e^{0 b} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.1.1.3.2

Multiply $0$ by $b$ .

$b^{3} e^{0} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.1.1.4

Anything raised to $0$ is $1$ .

$b^{3} \cdot 1 = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.1.1.5

Multiply $b^{3}$ by $1$ .

$b^{3} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.2

Simplify the right side.

Tap for more steps...

Step 10.2.2.1

Cancel the common factor of $e^{- b \sin (b \cdot 0)}$ .

Tap for more steps...

Step 10.2.2.1.1

Cancel the common factor.

$b^{3} = \frac{C}{e^{- b \sin (b \cdot 0)}} e^{- b \sin (b \cdot 0)}$

Step 10.2.2.1.2

Rewrite the expression.

$b^{3} = C$

Step 10.3

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

$b = \sqrt[3]{C}$

Step 11

Substitute $\sqrt[3]{C}$ for $C$ in $A = \frac{C}{e^{- b \sin (b r)}}$ and simplify.

Tap for more steps...

Step 11.1

Substitute $\sqrt[3]{C}$ for $b$ .

$A = \frac{C}{e^{- \sqrt[3]{C} \sin (\sqrt[3]{C} r)}}$