Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Subtract $x$ from both sides of the equation.

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

Step 1.1.2

Divide each term in $y \frac{d y}{d x} = 2 - x$ by $y$ and simplify.

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Step 1.1.2.1

Divide each term in $y \frac{d y}{d x} = 2 - x$ by $y$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $y$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 1.3

Multiply both sides by $y$ .

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

Step 1.4

Cancel the common factor of $y$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

$\frac{1}{2} y^{2} + C_{1} = 2 x - \frac{1}{2} x^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{2} x^{2} + 2 x + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

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

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

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

Step 3.2.2.1.3.1.2

Cancel the common factor.

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

Step 3.2.2.1.3.1.3

Rewrite the expression.

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

Step 3.2.2.1.3.2

Multiply $2$ by $2$ .

$y^{2} = - x^{2} + 4 x + 2 K$

Step 3.3

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

$y = \pm \sqrt{- x^{2} + 4 x + 2 K}$

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.

$y = \sqrt{- x^{2} + 4 x + 2 K}$

Step 3.4.2

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

$y = - \sqrt{- x^{2} + 4 x + 2 K}$

Step 3.4.3

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

$y = \sqrt{- x^{2} + 4 x + 2 K}$

$y = - \sqrt{- x^{2} + 4 x + 2 K}$

$y = \sqrt{- x^{2} + 4 x + 2 K}$

$y = - \sqrt{- x^{2} + 4 x + 2 K}$

$y = \sqrt{- x^{2} + 4 x + 2 K}$

$y = - \sqrt{- x^{2} + 4 x + 2 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt{- x^{2} + 4 x + K}$

$y = - \sqrt{- x^{2} + 4 x + K}$