Calculus Examples

$\frac{d y}{d x} = - \frac{x}{y}$ ; , $y (1) = 2$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

Cancel the common factor of $y$ .

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Step 1.2.2.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{x}{y}$

Step 1.2.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{x}{y}$

Step 1.2.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$y d y = - x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - x d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

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

Step 3.2.2.1.3.2

Cancel the common factor.

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

Step 3.2.2.1.3.3

Rewrite the expression.

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

Step 3.3

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

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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

Step 5

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

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

Step 6

Solve for $K$ .

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Step 6.1

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

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

Step 6.2

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

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

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

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

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

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Step 6.3.2.1.1

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

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Step 6.3.2.1.1.1

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

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

Step 6.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

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

Step 6.3.2.1.1.2.2

Rewrite the expression.

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

Step 6.3.2.1.2

Simplify each term.

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Step 6.3.2.1.2.1

One to any power is one.

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

Step 6.3.2.1.2.2

Multiply $- 1$ by $1$ .

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

Step 6.3.2.1.3

Simplify.

$- 1 + K = 2^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

$- 1 + K = 4$

Step 6.4

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

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Step 6.4.1

Add $1$ to both sides of the equation.

$K = 4 + 1$

Step 6.4.2

Add $4$ and $1$ .

$K = 5$

Step 7

Substitute $5$ for $K$ in $y = \sqrt{- x^{2} + K}$ and simplify.

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Step 7.1

Substitute $5$ for $K$ .

$y = \sqrt{- x^{2} + 5}$