Calculus Examples

$\frac{d y}{d x} = \frac{2 x}{y}$ and $y (1) = - 3$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{2 x}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{2 x}{y}$

Step 1.2.2

Rewrite the expression.

$y \frac{d y}{d x} = 2 x$

Step 1.3

Rewrite the equation.

$y d y = 2 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int 2 x d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C_{2})$ as $2 (\frac{1}{2}) x^{2} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{2} x^{2} + C_{2}$

Step 2.3.3.2.2.2

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = 1 x^{2} + C_{2}$

Step 2.3.3.2.3

Multiply $x^{2}$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = x^{2} + C_{2}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{2} = 2 x^{2} + 2 K$

Step 3.3

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

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

Step 3.4

Factor $2$ out of $2 x^{2} + 2 K$ .

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Step 3.4.1

Factor $2$ out of $2 x^{2}$ .

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

Step 3.4.2

Factor $2$ out of $2 K$ .

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

Step 3.4.3

Factor $2$ out of $2 (x^{2}) + 2 K$ .

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

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.

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

Step 3.5.2

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

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

Step 3.5.3

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

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

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

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

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

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

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

Step 4

Since $y$ is negative in the initial condition $(1, - 3)$ , only consider $y = - \sqrt{2 (x^{2} + K)}$ to find the $K$ . Substitute $1$ for $x$ and $- 3$ for $y$ .

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

Step 5

Solve for $K$ .

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Step 5.1

Rewrite the equation as $- \sqrt{2 (1^{2} + K)} = - 3$ .

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

Step 5.2

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

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

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 $\sqrt{2 (1^{2} + K)}$ as ${(2 (1^{2} + K))}^{\frac{1}{2}}$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

One to any power is one.

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

Step 5.3.2.1.2

Apply the distributive property.

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

Step 5.3.2.1.3

Simplify the expression.

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Step 5.3.2.1.3.1

Multiply $2$ by $1$ .

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

Step 5.3.2.1.3.2

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

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

Step 5.3.2.1.3.3

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

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

Step 5.3.2.1.3.4

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

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

Step 5.3.2.1.3.5

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

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Step 5.3.2.1.3.5.1

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

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

Step 5.3.2.1.3.5.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.3.5.2.1

Cancel the common factor.

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

Step 5.3.2.1.3.5.2.2

Rewrite the expression.

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

Step 5.3.2.1.4

Simplify.

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

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

$2 + 2 K = 9$

Step 5.4

Solve for $K$ .

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Step 5.4.1

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

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Step 5.4.1.1

Subtract $2$ from both sides of the equation.

$2 K = 9 - 2$

Step 5.4.1.2

Subtract $2$ from $9$ .

$2 K = 7$

Step 5.4.2

Divide each term in $2 K = 7$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 K = 7$ by $2$ .

$\frac{2 K}{2} = \frac{7}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{7}{2}$

Step 5.4.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{7}{2}$

Step 6

Substitute $\frac{7}{2}$ for $K$ in $y = - \sqrt{2 (x^{2} + K)}$ and simplify.

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Step 6.1

Substitute $\frac{7}{2}$ for $K$ .

$y = - \sqrt{2 (x^{2} + \frac{7}{2})}$

Step 6.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \sqrt{2 (x^{2} \cdot \frac{2}{2} + \frac{7}{2})}$

Step 6.3

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

$y = - \sqrt{2 (\frac{x^{2} \cdot 2}{2} + \frac{7}{2})}$

Step 6.4

Combine the numerators over the common denominator.

$y = - \sqrt{2 \frac{x^{2} \cdot 2 + 7}{2}}$

Step 6.5

Move $2$ to the left of $x^{2}$ .

$y = - \sqrt{2 \frac{2 x^{2} + 7}{2}}$

Step 6.6

Combine $2$ and $\frac{2 x^{2} + 7}{2}$ .

$y = - \sqrt{\frac{2 (2 x^{2} + 7)}{2}}$

Step 6.7

Reduce the expression by cancelling the common factors.

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Step 6.7.1

Reduce the expression $\frac{2 (2 x^{2} + 7)}{2}$ by cancelling the common factors.

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Step 6.7.1.1

Cancel the common factor.

$y = - \sqrt{\frac{2 (2 x^{2} + 7)}{2}}$

Step 6.7.1.2

Rewrite the expression.

$y = - \sqrt{\frac{2 x^{2} + 7}{1}}$

Step 6.7.2

Divide $2 x^{2} + 7$ by $1$ .

$y = - \sqrt{2 x^{2} + 7}$