Calculus Examples

$2 x y y' = 1 + y^{2}$

Step 1

Rewrite the differential equation.

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

Step 2

Separate the variables.

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Step 2.1

Divide each term in $2 x y \frac{d y}{d x} = 1 + y^{2}$ by $2 x y$ and simplify.

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Step 2.1.1

Divide each term in $2 x y \frac{d y}{d x} = 1 + y^{2}$ by $2 x y$ .

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Rewrite the expression.

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

Step 2.1.2.2

Cancel the common factor of $x$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Cancel the common factor of $y$ .

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Step 2.1.2.3.1

Cancel the common factor.

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

Step 2.1.2.3.2

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

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 2.1.3.1.1

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

$\frac{d y}{d x} = \frac{1}{2 x y} + \frac{y \cdot y}{2 x y}$

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $y$ out of $2 x y$ .

$\frac{d y}{d x} = \frac{1}{2 x y} + \frac{y \cdot y}{y (2 x)}$

Step 2.1.3.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{1}{2 x y} + \frac{y \cdot y}{y (2 x)}$

Step 2.1.3.1.2.3

Rewrite the expression.

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

Step 2.2

Factor.

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Step 2.2.1

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

$\frac{d y}{d x} = \frac{1}{2 x y} + \frac{y}{2 x} \cdot \frac{y}{y}$

Step 2.2.2

Multiply $\frac{y}{2 x}$ by $\frac{y}{y}$ .

$\frac{d y}{d x} = \frac{1}{2 x y} + \frac{y \cdot y}{2 x y}$

Step 2.2.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = \frac{1 + y \cdot y}{2 x y}$

Step 2.2.4

Multiply $y$ by $y$ .

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

Step 2.3

Regroup factors.

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

Step 2.4

Multiply both sides by $\frac{2 y}{1 + y^{2}}$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $\frac{1 + y^{2}}{2 y}$ by $\frac{1}{x}$ .

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

Step 2.5.2

Cancel the common factor of $2 y$ .

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Step 2.5.2.1

Factor $2 y$ out of $2 y x$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 2.5.3

Cancel the common factor of $1 + y^{2}$ .

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Step 2.5.3.1

Cancel the common factor.

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

Step 2.5.3.2

Rewrite the expression.

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

Step 2.6

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Integrate the left side.

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Step 3.2.1

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

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

Step 3.2.2

Let $u = 1 + y^{2}$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.2.2.1

Let $u = 1 + y^{2}$ . Find $\frac{d u}{d y}$ .

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Step 3.2.2.1.1

Differentiate $1 + y^{2}$ .

$\frac{d}{d y} [1 + y^{2}]$

Step 3.2.2.1.2

By the Sum Rule, the derivative of $1 + y^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [y^{2}]$ .

$\frac{d}{d y} [1] + \frac{d}{d y} [y^{2}]$

Step 3.2.2.1.3

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [y^{2}]$

Step 3.2.2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$0 + 2 y$

Step 3.2.2.1.5

Add $0$ and $2 y$ .

$2 y$

Step 3.2.2.2

Rewrite the problem using $u$ and $d u$ .

$2 \int \frac{1}{u} \cdot \frac{1}{2} d u = \int \frac{1}{x} d x$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$2 \int \frac{1}{u \cdot 2} d u = \int \frac{1}{x} d x$

Step 3.2.3.2

Move $2$ to the left of $u$ .

$2 \int \frac{1}{2 u} d u = \int \frac{1}{x} d x$

Step 3.2.4

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

$2 (\frac{1}{2} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 3.2.5

Simplify.

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Step 3.2.5.1

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

$\frac{2}{2} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 3.2.5.2

Cancel the common factor of $2$ .

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Step 3.2.5.2.1

Cancel the common factor.

$\frac{2}{2} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 3.2.5.2.2

Rewrite the expression.

$1 \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 3.2.5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 3.2.6

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| u |) + C_{1} = \int \frac{1}{x} d x$

Step 3.2.7

Replace all occurrences of $u$ with $1 + y^{2}$ .

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

Step 3.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 3.4

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

$\ln (| 1 + y^{2} |) = \ln (| x |) + C$

Step 4

Solve for $y$ .

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Step 4.1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| 1 + y^{2} |) - \ln (| x |) = C$

Step 4.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| 1 + y^{2} |}{| x |}) = C$

Step 4.3

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| 1 + y^{2} |}{| x |})} = e^{C}$

Step 4.4

Rewrite $\ln (\frac{| 1 + y^{2} |}{| x |}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = \frac{| 1 + y^{2} |}{| x |}$

Step 4.5

Solve for $y$ .

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Step 4.5.1

Rewrite the equation as $\frac{| 1 + y^{2} |}{| x |} = e^{C}$ .

$\frac{| 1 + y^{2} |}{| x |} = e^{C}$

Step 4.5.2

Multiply both sides by $| x |$ .

$\frac{| 1 + y^{2} |}{| x |} | x | = e^{C} | x |$

Step 4.5.3

Simplify the left side.

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Step 4.5.3.1

Cancel the common factor of $| x |$ .

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Step 4.5.3.1.1

Cancel the common factor.

$\frac{| 1 + y^{2} |}{| x |} | x | = e^{C} | x |$

Step 4.5.3.1.2

Rewrite the expression.

$| 1 + y^{2} | = e^{C} | x |$

Step 4.5.4

Solve for $y$ .

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Step 4.5.4.1

Reorder factors in $e^{C} | x |$ .

$| 1 + y^{2} | = | x | e^{C}$

Step 4.5.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$1 + y^{2} = \pm | x | e^{C}$

Step 4.5.4.3

Reorder factors in $\pm | x | e^{C}$ .

$1 + y^{2} = \pm e^{C} | x |$

Step 4.5.4.4

Subtract $1$ from both sides of the equation.

$y^{2} = \pm e^{C} | x | - 1$

Step 4.5.4.5

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

$y = \pm \sqrt{\pm e^{C} | x | - 1}$

Step 5

Group the constant terms together.

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Step 5.1

Simplify the constant of integration.

$y = \pm \sqrt{\pm C | x | - 1}$

Step 5.2

Combine constants with the plus or minus.

$y = \pm \sqrt{C | x | - 1}$