Calculus Examples

$3 y'' + y = 0$ , $y = \sin (k x)$

Step 1

Find $y'$ .

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\sin (k x))$

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 $f (x) = \sin (x)$ and $g (x) = k x$ .

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Step 1.3.1.1

To apply the Chain Rule, set $u$ as $k x$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [k x]$

Step 1.3.1.2

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

$\cos (u) \frac{d}{d x} [k x]$

Step 1.3.1.3

Replace all occurrences of $u$ with $k x$ .

$\cos (k x) \frac{d}{d x} [k x]$

Step 1.3.2

Differentiate.

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Step 1.3.2.1

Since $k$ is constant with respect to $x$ , the derivative of $k x$ with respect to $x$ is $k \frac{d}{d x} [x]$ .

$\cos (k x) (k \frac{d}{d x} [x])$

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 $n = 1$ .

$\cos (k x) (k \cdot 1)$

Step 1.3.2.3

Simplify the expression.

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Step 1.3.2.3.1

Multiply $k$ by $1$ .

$\cos (k x) k$

Step 1.3.2.3.2

Reorder the factors of $\cos (k x) k$ .

$k \cos (k x)$

Step 1.4

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

$y' = k \cos (k x)$

Step 2

Find $y''$ .

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Step 2.1

Set up the derivative.

$y'' = \frac{d}{d x} [k \cos (k x)]$

Step 2.2

Since $k$ is constant with respect to $x$ , the derivative of $k \cos (k x)$ with respect to $x$ is $k \frac{d}{d x} [\cos (k x)]$ .

$y'' = k \frac{d}{d x} [\cos (k x)]$

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 $f (x) = \cos (x)$ and $g (x) = k x$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $k x$ .

$y'' = k (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [k x])$

Step 2.3.2

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

$y'' = k (- \sin (u) \frac{d}{d x} [k x])$

Step 2.3.3

Replace all occurrences of $u$ with $k x$ .

$y'' = k (- \sin (k x) \frac{d}{d x} [k x])$

Step 2.4

Since $k$ is constant with respect to $x$ , the derivative of $k x$ with respect to $x$ is $k \frac{d}{d x} [x]$ .

$y'' = k (- \sin (k x) (k \frac{d}{d x} [x]))$

Step 2.5

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

$y'' = k^{1} k (- \sin (k x) (\frac{d}{d x} [x]))$

Step 2.6

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

$y'' = k^{1} k^{1} (- \sin (k x) (\frac{d}{d x} [x]))$

Step 2.7

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

$y'' = k^{1 + 1} (- \sin (k x) (\frac{d}{d x} [x]))$

Step 2.8

Add $1$ and $1$ .

$y'' = k^{2} (- \sin (k x) (\frac{d}{d x} [x]))$

Step 2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y'' = k^{2} (- \sin (k x) \cdot 1)$

Step 2.10

Multiply $- 1$ by $1$ .

$y'' = k^{2} (- \sin (k x))$

Step 2.11

Reorder the factors of $k^{2} (- \sin (k x))$ .

$y'' = - k^{2} \sin (k x)$

Step 3

Substitute into the given differential equation.

$3 (- k^{2} \sin (k x)) + y = 0$

Step 4

Substitute $y$ for $\sin (k x)$ .

$3 (- k^{2} y) + y = 0$

Step 5

Solve for $k$ .

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Step 5.1

Multiply $- 1$ by $3$ .

$- 3 k^{2} y + y = 0$

Step 5.2

Subtract $y$ from both sides of the equation.

$- 3 k^{2} y = - y$

Step 5.3

Divide each term in $- 3 k^{2} y = - y$ by $- 3 y$ and simplify.

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Step 5.3.1

Divide each term in $- 3 k^{2} y = - y$ by $- 3 y$ .

$\frac{- 3 k^{2} y}{- 3 y} = \frac{- y}{- 3 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 k^{2} y}{- 3 y} = \frac{- y}{- 3 y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{k^{2} y}{y} = \frac{- y}{- 3 y}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{k^{2} y}{y} = \frac{- y}{- 3 y}$

Step 5.3.2.2.2

Divide $k^{2}$ by $1$ .

$k^{2} = \frac{- y}{- 3 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $y$ .

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Step 5.3.3.1.1

Cancel the common factor.

$k^{2} = \frac{- y}{- 3 y}$

Step 5.3.3.1.2

Rewrite the expression.

$k^{2} = \frac{- 1}{- 3}$

Step 5.3.3.2

Dividing two negative values results in a positive value.

$k^{2} = \frac{1}{3}$

Step 5.4

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

$k = \pm \sqrt{\frac{1}{3}}$

Step 5.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 5.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$k = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 5.5.2

Any root of $1$ is $1$ .

$k = \pm \frac{1}{\sqrt{3}}$

Step 5.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$k = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.5.4

Combine and simplify the denominator.

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Step 5.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$k = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$k = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$k = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.5.4.4

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

$k = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.5.4.5

Add $1$ and $1$ .

$k = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$k = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.5.4.6.3

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

$k = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.5.4.6.4

Cancel the common factor of $2$ .

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Step 5.5.4.6.4.1

Cancel the common factor.

$k = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.5.4.6.4.2

Rewrite the expression.

$k = \pm \frac{\sqrt{3}}{3^{1}}$

Step 5.5.4.6.5

Evaluate the exponent.

$k = \pm \frac{\sqrt{3}}{3}$

Step 5.6

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

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$k = \frac{\sqrt{3}}{3}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$k = - \frac{\sqrt{3}}{3}$

Step 5.6.3

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

$k = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$k = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Decimal Form:

$k = 0.57735026 \dots, - 0.57735026 \dots$

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