Calculus Examples

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

Step 1

Rewrite the differential equation.

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

Step 2

Separate the variables.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $x$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Rewrite the expression.

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

Step 2.1.2.2

Cancel the common factor of $y$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

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

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

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

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Step 2.1.3.1.1.1

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

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

Step 2.1.3.1.1.2

Cancel the common factors.

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Step 2.1.3.1.1.2.1

Factor $x$ out of $x y$ .

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

Step 2.1.3.1.1.2.2

Cancel the common factor.

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

Step 2.1.3.1.1.2.3

Rewrite the expression.

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

Step 2.1.3.1.2

Move the negative in front of the fraction.

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

Step 2.2

Factor.

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Step 2.2.1

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

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

Step 2.2.2

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

Reorder the factors of $y x$ .

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.2.4.2

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 2.2.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 2.3

Regroup factors.

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

Step 2.4

Multiply both sides by $y$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $\frac{(1 + x) (1 - x)}{x}$ by $\frac{1}{y}$ .

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

Step 2.5.2

Cancel the common factor of $y$ .

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Step 2.5.2.1

Factor $y$ out of $x y$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 2.6

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

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

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Reorder $x$ and $1$ .

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

Step 3.3.4.2

Reorder $x$ and $- 1$ .

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

Step 3.3.4.3

Multiply $1$ by $1$ .

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

Step 3.3.4.4

Multiply $- 1$ by $1$ .

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

Step 3.3.4.5

Multiply $x$ by $1$ .

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

Step 3.3.5

Factor out negative.

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

Step 3.3.6

Raise $x$ to the power of $1$ .

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

Step 3.3.7

Raise $x$ to the power of $1$ .

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

Step 3.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.9

Add $1$ and $1$ .

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

Step 3.3.10

Add $- x$ and $x$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1 + 0 - x^{2}}{x} d x$

Step 3.3.11

Simplify the expression.

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Step 3.3.11.1

Subtract $x^{2}$ from $0$ .

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

Step 3.3.11.2

Reorder $1$ and $- x^{2}$ .

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

Step 3.3.12

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

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Step 3.3.12.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$x^{2}$

$0 x$

$1$

Step 3.3.12.2

Divide the highest order term in the dividend $- x^{2}$ by the highest order term in divisor $x$ .

				-	$x$
	$x$	+	$0$	-	$x^{2}$	+	$0 x$	+	$1$

Step 3.3.12.3

Multiply the new quotient term by the divisor.

			-	$x$
$x$	+	$0$	-	$x^{2}$	+	$0 x$	+	$1$
			-	$x^{2}$	+	$0$

Step 3.3.12.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} + 0$

			-	$x$
$x$	+	$0$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$0$

Step 3.3.12.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$x$
$x$	+	$0$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$0$
						$0$

Step 3.3.12.6

Pull the next term from the original dividend down into the current dividend.

			-	$x$
$x$	+	$0$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$0$
						$0$	+	$1$

Step 3.3.12.7

The final answer is the quotient plus the remainder over the divisor.

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

Step 3.3.13

Split the single integral into multiple integrals.

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

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

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

Step 3.3.17

Simplify.

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Step 3.3.17.1

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

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

Step 3.3.17.2

Simplify.

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{2} x^{2} + \ln (| x |) + C_{4}$

Step 3.4

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

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

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

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

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

Apply the distributive property.

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

Step 4.2.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.1

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

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

Step 4.2.2.1.3.2

Cancel the common factor.

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

Step 4.2.2.1.3.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

$y = \pm \sqrt{- x^{2} + \ln (x^{2}) + 2 C}$

Step 4.6

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

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Step 4.6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.6.2

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

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

Step 4.6.3

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

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

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

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

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

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

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

Step 5

Simplify the constant of integration.

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

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