Calculus Examples

$\frac{d y}{d x} = \frac{x \sin (x)}{y}$ , $y (0) = - 1$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$y d y = x \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int x \sin (x) d x$

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int x \sin (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (x)$ .

$\frac{1}{2} y^{2} + C_{1} = x (- \cos (x)) - \int - \cos (x) d x$

Step 2.3.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = x (- \cos (x)) - - \int \cos (x) d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} y^{2} + C_{1} = x (- \cos (x)) + 1 \int \cos (x) d x$

Step 2.3.3.2

Multiply $\int \cos (x) d x$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = x (- \cos (x)) + \int \cos (x) d x$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = x (- \cos (x)) + \sin (x) + C_{2}$

Step 2.3.5

Rewrite $x (- \cos (x)) + \sin (x) + C_{2}$ as $- x \cos (x) + \sin (x) + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - x \cos (x) + \sin (x) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - x \cos (x) + \sin (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- x \cos (x) + \sin (x) + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (- x \cos (x) + \sin (x) + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (- x \cos (x) + \sin (x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- x \cos (x) + \sin (x) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- x \cos (x) + \sin (x) + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{2} = 2 (- x \cos (x)) + 2 \sin (x) + 2 K$

Step 3.2.2.1.2

Multiply $- 1$ by $2$ .

$y^{2} = - 2 x \cos (x) + 2 \sin (x) + 2 K$

Step 3.3

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

$y = \pm \sqrt{- 2 x \cos (x) + 2 \sin (x) + 2 K}$

Step 3.4

Factor $2$ out of $- 2 x \cos (x) + 2 \sin (x) + 2 K$ .

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Step 3.4.1

Factor $2$ out of $- 2 x \cos (x)$ .

$y = \pm \sqrt{2 (- x \cos (x)) + 2 \sin (x) + 2 K}$

Step 3.4.2

Factor $2$ out of $2 \sin (x)$ .

$y = \pm \sqrt{2 (- x \cos (x)) + 2 (\sin (x)) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (- x \cos (x)) + 2 (\sin (x)) + 2 K}$

Step 3.4.4

Factor $2$ out of $2 (- x \cos (x)) + 2 (\sin (x))$ .

$y = \pm \sqrt{2 (- x \cos (x) + \sin (x)) + 2 K}$

Step 3.4.5

Factor $2$ out of $2 (- x \cos (x) + \sin (x)) + 2 K$ .

$y = \pm \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

Step 3.5

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

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Step 3.5.1

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

$y = \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

Step 3.5.2

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

$y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

Step 3.5.3

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

$y = \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

$y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

$y = \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

$y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

$y = \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

$y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$

Step 4

Since $y$ is negative in the initial condition $(0, - 1)$ , only consider $y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$ to find the $K$ . Substitute $0$ for $x$ and $- 1$ for $y$ .

$- 1 = - \sqrt{2 (0 \cos (0) + \sin (0) + K)}$

Step 5

Solve for $K$ .

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Step 5.1

Rewrite the equation as $- \sqrt{2 (0 \cos (0) + \sin (0) + K)} = - 1$ .

$- \sqrt{2 (0 \cos (0) + \sin (0) + K)} = - 1$

Step 5.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{2 (0 \cos (0) + \sin (0) + K)})}^{2} = {(- 1)}^{2}$

Step 5.3

Simplify each side of the equation.

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Step 5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (0 \cos (0) + \sin (0) + K)}$ as ${(2 (0 \cos (0) + \sin (0) + K))}^{\frac{1}{2}}$ .

${(- {(2 (0 \cos (0) + \sin (0) + K))}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify ${(- {(2 (0 \cos (0) + \sin (0) + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1

Simplify each term.

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Step 5.3.2.1.1.1

The exact value of $\cos (0)$ is $1$ .

${(- {(2 (0 \cdot 1 + \sin (0) + K))}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.1.2

Multiply $0$ by $1$ .

${(- {(2 (0 + \sin (0) + K))}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.1.3

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

${(- {(2 (0 + 0 + K))}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.2

Simplify by adding terms.

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Step 5.3.2.1.2.1

Combine the opposite terms in $0 + 0 + K$ .

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Step 5.3.2.1.2.1.1

Add $0$ and $0$ .

${(- {(2 (0 + K))}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.2.1.2

Add $0$ and $K$ .

${(- {(2 K)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.2.2

Apply the product rule to $2 K$ .

${(- (2^{\frac{1}{2}} K^{\frac{1}{2}}))}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.2.1.3.1

Apply the product rule to $- 2^{\frac{1}{2}} K^{\frac{1}{2}}$ .

${(- 2^{\frac{1}{2}})}^{2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.3.2

Apply the product rule to $- 2^{\frac{1}{2}}$ .

${(- 1)}^{2} {(2^{\frac{1}{2}})}^{2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.4

Raise $- 1$ to the power of $2$ .

$1 {(2^{\frac{1}{2}})}^{2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.5

Multiply ${(2^{\frac{1}{2}})}^{2}$ by $1$ .

${(2^{\frac{1}{2}})}^{2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.6

Multiply the exponents in ${(2^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.6.1

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

$2^{\frac{1}{2} \cdot 2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.6.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.6.2.1

Cancel the common factor.

$2^{\frac{1}{2} \cdot 2} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.6.2.2

Rewrite the expression.

$2^{1} {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.7

Evaluate the exponent.

$2 {(K^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 5.3.2.1.8

Multiply the exponents in ${(K^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.8.1

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

$2 K^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 5.3.2.1.8.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.8.2.1

Cancel the common factor.

$2 K^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 5.3.2.1.8.2.2

Rewrite the expression.

$2 K^{1} = {(- 1)}^{2}$

Step 5.3.2.1.9

Simplify.

$2 K = {(- 1)}^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Raise $- 1$ to the power of $2$ .

$2 K = 1$

Step 5.4

Divide each term in $2 K = 1$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 K = 1$ by $2$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $K$ by $1$ .

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

Step 6

Substitute $\frac{1}{2}$ for $K$ in $y = - \sqrt{2 (- x \cos (x) + \sin (x) + K)}$ and simplify.

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Step 6.1

Substitute $\frac{1}{2}$ for $K$ .

$y = - \sqrt{2 (- x \cos (x) + \sin (x) + \frac{1}{2})}$