Calculus Examples

$\frac{d y}{d x} = \frac{2 x + \sec (x) \tan (x)}{2 y}$ , $y (0) = - 3$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{2 x + \sec (x) \tan (x)}{2 y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $2 y$ .

$y \frac{d y}{d x} = y \frac{2 x + \sec (x) \tan (x)}{y \cdot 2}$

Step 1.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{2 x + \sec (x) \tan (x)}{y \cdot 2}$

Step 1.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{2 x + \sec (x) \tan (x)}{2}$

Step 1.3

Rewrite the equation.

$y d y = \frac{2 x + \sec (x) \tan (x)}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{2 x + \sec (x) \tan (x)}{2} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} (2 (\frac{1}{2} x^{2} + C_{2}) + \sec (x) + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} (2 (\frac{x^{2}}{2} + C_{2}) + \sec (x) + C_{3})$

Step 2.3.6.2

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} (x^{2} + \sec (x)) + C_{4}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{2} (x^{2} + \sec (x)) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{2} (x^{2} + \sec (x)) + K)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{2} (x^{2} + \sec (x)) + K)$

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.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{2} (x^{2} + \sec (x)) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{2} (x^{2} + \sec (x)) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{2} (x^{2} + \sec (x)) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$y^{2} = 2 (\frac{1}{2} x^{2} + \frac{1}{2} \sec (x) + K)$

Step 3.2.2.1.1.2

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

$y^{2} = 2 (\frac{x^{2}}{2} + \frac{1}{2} \sec (x) + K)$

Step 3.2.2.1.1.3

Combine $\frac{1}{2}$ and $\sec (x)$ .

$y^{2} = 2 (\frac{x^{2}}{2} + \frac{\sec (x)}{2} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{2}}{2} + 2 \frac{\sec (x)}{2} + 2 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

$y^{2} = 2 \frac{x^{2}}{2} + 2 \frac{\sec (x)}{2} + 2 K$

Step 3.2.2.1.3.1.2

Rewrite the expression.

$y^{2} = x^{2} + 2 \frac{\sec (x)}{2} + 2 K$

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

$y^{2} = x^{2} + 2 \frac{\sec (x)}{2} + 2 K$

Step 3.2.2.1.3.2.2

Rewrite the expression.

$y^{2} = x^{2} + \sec (x) + 2 K$

Step 3.3

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

$y = \pm \sqrt{x^{2} + \sec (x) + 2 K}$

Step 3.4

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

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Step 3.4.1

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

$y = \sqrt{x^{2} + \sec (x) + 2 K}$

Step 3.4.2

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

$y = - \sqrt{x^{2} + \sec (x) + 2 K}$

Step 3.4.3

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

$y = \sqrt{x^{2} + \sec (x) + 2 K}$

$y = - \sqrt{x^{2} + \sec (x) + 2 K}$

$y = \sqrt{x^{2} + \sec (x) + 2 K}$

$y = - \sqrt{x^{2} + \sec (x) + 2 K}$

$y = \sqrt{x^{2} + \sec (x) + 2 K}$

$y = - \sqrt{x^{2} + \sec (x) + 2 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt{x^{2} + \sec (x) + K}$

$y = - \sqrt{x^{2} + \sec (x) + K}$

Step 5

Since $y$ is negative in the initial condition $(0, - 3)$ , only consider $y = - \sqrt{x^{2} + \sec (x) + K}$ to find the $K$ . Substitute $0$ for $x$ and $- 3$ for $y$ .

$- 3 = - \sqrt{0^{2} + \sec (0) + K}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $- \sqrt{0^{2} + \sec (0) + K} = - 3$ .

$- \sqrt{0^{2} + \sec (0) + K} = - 3$

Step 6.2

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

${(- \sqrt{0^{2} + \sec (0) + K})}^{2} = {(- 3)}^{2}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

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

${(- {(0^{2} + \sec (0) + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

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

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Step 6.3.2.1.1

Simplify each term.

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Step 6.3.2.1.1.1

Raising $0$ to any positive power yields $0$ .

${(- {(0 + \sec (0) + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.1.2

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

${(- {(0 + 1 + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.2

Simplify the expression.

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Step 6.3.2.1.2.1

Add $0$ and $1$ .

${(- {(1 + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.2.2

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

${(- 1)}^{2} {({(1 + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.2.3

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

$1 {({(1 + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.2.4

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

${({(1 + K)}^{\frac{1}{2}})}^{2} = {(- 3)}^{2}$

Step 6.3.2.1.2.5

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

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Step 6.3.2.1.2.5.1

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

${(1 + K)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 6.3.2.1.2.5.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.2.5.2.1

Cancel the common factor.

${(1 + K)}^{\frac{1}{2} \cdot 2} = {(- 3)}^{2}$

Step 6.3.2.1.2.5.2.2

Rewrite the expression.

${(1 + K)}^{1} = {(- 3)}^{2}$

Step 6.3.2.1.3

Simplify.

$1 + K = {(- 3)}^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Raise $- 3$ to the power of $2$ .

$1 + K = 9$

Step 6.4

Move all terms not containing $K$ to the right side of the equation.

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Step 6.4.1

Subtract $1$ from both sides of the equation.

$K = 9 - 1$

Step 6.4.2

Subtract $1$ from $9$ .

$K = 8$

Step 7

Substitute $8$ for $K$ in $y = - \sqrt{x^{2} + \sec (x) + K}$ and simplify.

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Step 7.1

Substitute $8$ for $K$ .

$y = - \sqrt{x^{2} + \sec (x) + 8}$