Calculus Examples

$(1 + y^{2}) d x + x y d y = 0$

Step 1

Subtract $(1 + y^{2}) d x$ from both sides of the equation.

$x y d y = - (1 + y^{2}) d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Combine $\frac{1}{1 + y^{2}}$ and $y$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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Step 3.4.1

Move the leading negative in $- \frac{1}{x (1 + y^{2})}$ into the numerator.

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

Step 3.4.2

Factor $1 + y^{2}$ out of $x (1 + y^{2})$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 3.5

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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 4.2.1.1

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

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Step 4.2.1.1.1

Differentiate $1 + y^{2}$ .

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

Step 4.2.1.1.2

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 4.2.1.1.3

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 4.2.1.1.4

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 4.2.1.1.5

Add $0$ and $2 y$ .

$2 y$

Step 4.2.1.2

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Simplify.

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

Step 4.2.6

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $2 (\frac{1}{2} \ln (| 1 + y^{2} |))$ .

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Step 5.2.1.1.1

Combine $\frac{1}{2}$ and $\ln (| 1 + y^{2} |)$ .

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 (- \ln (| x |) + C)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Multiply $- 1$ by $2$ .

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

Step 5.3

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

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

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $\ln (| 1 + y^{2} |) + 2 \ln (| x |)$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

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

Step 5.4.1.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

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

Step 5.4.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 5.4.1.3

Reorder factors in $\ln (| 1 + y^{2} | x^{2})$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Solve for $y$ .

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Step 5.7.1

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

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

Step 5.7.2

Divide each term in $x^{2} | 1 + y^{2} | = e^{2 C}$ by $x^{2}$ and simplify.

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Step 5.7.2.1

Divide each term in $x^{2} | 1 + y^{2} | = e^{2 C}$ by $x^{2}$ .

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

Step 5.7.2.2

Simplify the left side.

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Step 5.7.2.2.1

Cancel the common factor of $x^{2}$ .

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Step 5.7.2.2.1.1

Cancel the common factor.

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

Step 5.7.2.2.1.2

Divide $| 1 + y^{2} |$ by $1$ .

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

Step 5.7.3

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

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

Step 5.7.4

Subtract $1$ from both sides of the equation.

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

Step 5.7.5

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

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

Step 6

Group the constant terms together.

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Step 6.1

Simplify the constant of integration.

$y = \pm \sqrt{\pm \frac{C}{x^{2}} - 1}$

Step 6.2

Combine constants with the plus or minus.

$y = \pm \sqrt{C \frac{1}{x^{2}} - 1}$