Calculus Examples

$4 y'' = y$ , $y = e^{r x}$

Step 1

Find $y'$ .

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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.

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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 $g (x) = r x$ .

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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.

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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.

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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 $y''$ .

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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 $g (x) = r x$ .

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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 $r$ .

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Step 5.1

Divide each term in $4 r^{2} y = y$ by $4 y$ and simplify.

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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.

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Step 5.1.2.1

Cancel the common factor of $4$ .

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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 $y$ .

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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.

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Step 5.1.3.1

Cancel the common factor of $y$ .

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

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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.

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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.

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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.

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

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