Calculus Examples

$x (1 - y) \frac{d y}{d x} + (1 - x) y = 0$

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

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$(x \cdot 1 + x (- y)) \frac{d y}{d x} + (1 - x) y = 0$

Step 1.1.1.2

Multiply $x$ by $1$ .

$(x + x (- y)) \frac{d y}{d x} + (1 - x) y = 0$

Step 1.1.1.3

Rewrite using the commutative property of multiplication.

$(x - x y) \frac{d y}{d x} + (1 - x) y = 0$

Step 1.1.1.4

Apply the distributive property.

$x \frac{d y}{d x} - x y \frac{d y}{d x} + (1 - x) y = 0$

Step 1.1.1.5

Apply the distributive property.

$x \frac{d y}{d x} - x y \frac{d y}{d x} + 1 y - x y = 0$

Step 1.1.1.6

Multiply $y$ by $1$ .

$x \frac{d y}{d x} - x y \frac{d y}{d x} + y - x y = 0$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Subtract $y$ from both sides of the equation.

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

Step 1.1.2.2

Add $x y$ to both sides of the equation.

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

Step 1.1.3

Factor $x \frac{d y}{d x}$ out of $x \frac{d y}{d x} - x y \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $x \frac{d y}{d x}$ out of $x \frac{d y}{d x}$ .

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

Step 1.1.3.2

Factor $x \frac{d y}{d x}$ out of $- x y \frac{d y}{d x}$ .

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

Step 1.1.3.3

Factor $x \frac{d y}{d x}$ out of $x \frac{d y}{d x} \cdot 1 + x \frac{d y}{d x} (- 1 y)$ .

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

Step 1.1.4

Rewrite $- 1 y$ as $- y$ .

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

Step 1.1.5

Divide each term in $x \frac{d y}{d x} (1 - y) = - y + x y$ by $x (1 - y)$ and simplify.

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Step 1.1.5.1

Divide each term in $x \frac{d y}{d x} (1 - y) = - y + x y$ by $x (1 - y)$ .

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

Step 1.1.5.2

Simplify the left side.

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Step 1.1.5.2.1

Cancel the common factor of $x$ .

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Step 1.1.5.2.1.1

Cancel the common factor.

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

Step 1.1.5.2.1.2

Rewrite the expression.

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

Step 1.1.5.2.2

Cancel the common factor of $1 - y$ .

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Step 1.1.5.2.2.1

Cancel the common factor.

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

Step 1.1.5.2.2.2

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

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

Step 1.1.5.3

Simplify the right side.

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Step 1.1.5.3.1

Simplify each term.

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Step 1.1.5.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.5.3.1.2

Cancel the common factor of $x$ .

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Step 1.1.5.3.1.2.1

Cancel the common factor.

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

Step 1.1.5.3.1.2.2

Rewrite the expression.

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

Step 1.1.5.3.2

To write $\frac{y}{1 - y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.1.5.3.3

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

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Step 1.1.5.3.3.1

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

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

Step 1.1.5.3.3.2

Reorder the factors of $(1 - y) x$ .

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

Step 1.1.5.3.4

Combine the numerators over the common denominator.

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

Step 1.1.5.3.5

Factor $y$ out of $- y + y x$ .

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Step 1.1.5.3.5.1

Factor $y$ out of $- y$ .

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

Step 1.1.5.3.5.2

Factor $y$ out of $y x$ .

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

Step 1.1.5.3.5.3

Factor $y$ out of $y \cdot - 1 + y (x)$ .

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

Step 1.1.5.3.6

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.1.5.3.7

Factor $- 1$ out of $x$ .

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

Step 1.1.5.3.8

Factor $- 1$ out of $- 1 (1) - 1 (- x)$ .

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

Step 1.1.5.3.9

Move the negative in front of the fraction.

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{1 - y}{y}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2

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

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

Step 1.4.3

Cancel the common factor of $1 - y$ .

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Step 1.4.3.1

Move the leading negative in $- \frac{1 - y}{y}$ into the numerator.

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

Step 1.4.3.2

Factor $1 - y$ out of $- (1 - y)$ .

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

Step 1.4.3.3

Factor $1 - y$ out of $(1 - y) x$ .

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

Step 1.4.3.4

Cancel the common factor.

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

Step 1.4.3.5

Rewrite the expression.

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

Step 1.4.4

Cancel the common factor of $y$ .

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Step 1.4.4.1

Cancel the common factor.

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

Step 1.4.4.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Cancel the common factor of $y$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Divide $- 1$ by $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Apply the constant rule.

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Reorder terms.

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

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.

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

Step 2.3.2

Reorder $1$ and $- x$ .

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

Step 2.3.3

Divide $- x + 1$ by $x$ .

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Step 2.3.3.1

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

$x$

$0$

$x$

$1$

Step 2.3.3.2

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

				-	$1$
	$x$	+	$0$	-	$x$	+	$1$

Step 2.3.3.3

Multiply the new quotient term by the divisor.

			-	$1$
$x$	+	$0$	-	$x$	+	$1$
			-	$x$	+	$0$

Step 2.3.3.4

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

			-	$1$
$x$	+	$0$	-	$x$	+	$1$
			+	$x$	-	$0$

Step 2.3.3.5

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

			-	$1$
$x$	+	$0$	-	$x$	+	$1$
			+	$x$	-	$0$
					+	$1$

Step 2.3.3.6

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

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

Step 2.3.4

Split the single integral into multiple integrals.

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

Step 2.3.5

Apply the constant rule.

$- y + \ln (| y |) + C_{3} = - (- x + C_{4} + \int \frac{1}{x} d x)$

Step 2.3.6

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

$- y + \ln (| y |) + C_{3} = - (- x + C_{4} + \ln (| x |) + C_{5})$

Step 2.3.7

Simplify.

$- y + \ln (| y |) + C_{3} = - (- x + \ln (| x |)) + C_{6}$

Step 2.4

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

$- y + \ln (| y |) = - (- x + \ln (| x |)) + C$