Calculus Examples

$4 y'' = y$ ,

$y = e^{r x}$

Step 1

Find

$y'$ .

Tap for more steps...

Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{r x})$

Step 1.2

The derivative of

$y$ with respect to

$x$ is

$y'$ .

$y'$

Step 1.3

Differentiate the right side of the equation.

Tap for more steps...

Step 1.3.1

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = e^{x}$ and

$g (x) = r x$ .

Tap for more steps...

Step 1.3.1.1

To apply the Chain Rule, set

$u$ as

$r x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [r x]$

Step 1.3.1.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u} [a^{u}]$ is

$a^{u} \ln (a)$ where

$a$ =

$e$ .

$e^{u} \frac{d}{d x} [r x]$

Step 1.3.1.3

Replace all occurrences of

$u$ with

$r x$ .

$e^{r x} \frac{d}{d x} [r x]$

Step 1.3.2

Differentiate.

Tap for more steps...

Step 1.3.2.1

Since

$r$ is constant with respect to

$x$ , the derivative of

$r x$ with respect to

$x$ is

$r \frac{d}{d x} [x]$ .

$e^{r x} (r \frac{d}{d x} [x])$

Step 1.3.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$e^{r x} (r \cdot 1)$

Step 1.3.2.3

Simplify the expression.

Tap for more steps...

Step 1.3.2.3.1

Multiply

$r$ by

$1$ .

$e^{r x} r$

Step 1.3.2.3.2

Reorder factors in

$e^{r x} r$ .

$r e^{r x}$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$y' = r e^{r x}$

Step 2

Find

$y''$ .

Tap for more steps...

Step 2.1

Set up the derivative.

$y'' = \frac{d}{d x} [r e^{r x}]$

Step 2.2

Since

$r$ is constant with respect to

$x$ , the derivative of

$r e^{r x}$ with respect to

$x$ is

$r \frac{d}{d x} [e^{r x}]$ .

$y'' = r \frac{d}{d x} [e^{r x}]$

Step 2.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = e^{x}$ and

$g (x) = r x$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set

$u$ as

$r x$ .

$y'' = r (\frac{d}{d u} [e^{u}] \frac{d}{d x} [r x])$

Step 2.3.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u} [a^{u}]$ is

$a^{u} \ln (a)$ where

$a$ =

$e$ .

$y'' = r (e^{u} \frac{d}{d x} [r x])$

Step 2.3.3

Replace all occurrences of

$u$ with

$r x$ .

$y'' = r (e^{r x} \frac{d}{d x} [r x])$

Step 2.4

Since

$r$ is constant with respect to

$x$ , the derivative of

$r x$ with respect to

$x$ is

$r \frac{d}{d x} [x]$ .

$y'' = r (e^{r x} (r \frac{d}{d x} [x]))$

Step 2.5

Raise

$r$ to the power of

$1$ .

$y'' = r^{1} r (e^{r x} (\frac{d}{d x} [x]))$

Step 2.6

Raise

$r$ to the power of

$1$ .

$y'' = r^{1} r^{1} (e^{r x} (\frac{d}{d x} [x]))$

Step 2.7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y'' = r^{1 + 1} (e^{r x} (\frac{d}{d x} [x]))$

Step 2.8

Add

$1$ and

$1$ .

$y'' = r^{2} (e^{r x} (\frac{d}{d x} [x]))$

Step 2.9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$y'' = r^{2} (e^{r x} \cdot 1)$

Step 2.10

Multiply

$e^{r x}$ by

$1$ .

$y'' = r^{2} e^{r x}$

Step 3

Substitute into the given differential equation.

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

Step 4

Substitute

$y$ for

$e^{r x}$ .

$4 (r^{2} y) = y$

Step 5

Solve for

$r$ .

Tap for more steps...

Step 5.1

Divide each term in

$4 r^{2} y = y$ by

$4 y$ and simplify.

Tap for more steps...

Step 5.1.1

Divide each term in

$4 r^{2} y = y$ by

$4 y$ .

$\frac{4 r^{2} y}{4 y} = \frac{y}{4 y}$

Step 5.1.2

Simplify the left side.

Tap for more steps...

Step 5.1.2.1

Cancel the common factor of

$4$ .

Tap for more steps...

Step 5.1.2.1.1

Cancel the common factor.

$\frac{4 r^{2} y}{4 y} = \frac{y}{4 y}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{r^{2} y}{y} = \frac{y}{4 y}$

Step 5.1.2.2

Cancel the common factor of

$y$ .

Tap for more steps...

Step 5.1.2.2.1

Cancel the common factor.

$\frac{r^{2} y}{y} = \frac{y}{4 y}$

Step 5.1.2.2.2

Divide

$r^{2}$ by

$1$ .

$r^{2} = \frac{y}{4 y}$

Step 5.1.3

Simplify the right side.

Tap for more steps...

Step 5.1.3.1

Cancel the common factor of

$y$ .

Tap for more steps...

Step 5.1.3.1.1

Cancel the common factor.

$r^{2} = \frac{y}{4 y}$

Step 5.1.3.1.2

Rewrite the expression.

$r^{2} = \frac{1}{4}$

Step 5.2

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

$r = \pm \sqrt{\frac{1}{4}}$

Step 5.3

Simplify

$\pm \sqrt{\frac{1}{4}}$ .

Tap for more steps...

Step 5.3.1

Rewrite

$\sqrt{\frac{1}{4}}$ as

$\frac{\sqrt{1}}{\sqrt{4}}$ .

$r = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.3.2

Any root of

$1$ is

$1$ .

$r = \pm \frac{1}{\sqrt{4}}$

Step 5.3.3

Simplify the denominator.

Tap for more steps...

Step 5.3.3.1

Rewrite

$4$ as

$2^{2}$ .

$r = \pm \frac{1}{\sqrt{2^{2}}}$

Step 5.3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$r = \pm \frac{1}{2}$

Step 5.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 5.4.1

First, use the positive value of the

$\pm$ to find the first solution.

$r = \frac{1}{2}$

Step 5.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

$r = - \frac{1}{2}$

Step 5.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{1}{2}, - \frac{1}{2}$

Enter YOUR Problem