Calculus Examples

$(1 - x) d y - (y d) x = 0$

Step 1

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

$(1 - x) d y = y d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $1 - x$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.1.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

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Step 4.3.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 4.3.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 4.3.1.2

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

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

Step 4.3.2

Split the fraction into multiple fractions.

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

Step 4.3.3

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

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

Step 4.3.4

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

$\ln (| y |) + C_{1} = - (\ln (| u |) + C_{2})$

Step 4.3.5

Simplify.

$\ln (| y |) + C_{1} = - \ln (| u |) + C_{2}$

Step 4.3.6

Replace all occurrences of $u$ with $1 - x$ .

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

Step 4.4

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

$\ln (| y |) = - \ln (| 1 - x |) + C$

Step 5

Solve for $y$ .

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Step 5.1

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

$\ln (| y |) + \ln (| 1 - x |) = C$

Step 5.2

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

$\ln (| y | | 1 - x |) = C$

Step 5.3

To multiply absolute values, multiply the terms inside each absolute value.

$\ln (| y (1 - x) |) = C$

Step 5.4

Apply the distributive property.

$\ln (| y \cdot 1 + y (- x) |) = C$

Step 5.5

Simplify the expression.

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Step 5.5.1

Multiply $y$ by $1$ .

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

Step 5.5.2

Rewrite using the commutative property of multiplication.

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

Step 5.6

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

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

Step 5.7

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

$e^{C} = | y - y x |$

Step 5.8

Solve for $y$ .

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Step 5.8.1

Rewrite the equation as $| y - y x | = e^{C}$ .

$| y - y x | = e^{C}$

Step 5.8.2

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

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

Step 5.8.3

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

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Step 5.8.3.1

Factor $y$ out of $y^{1}$ .

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

Step 5.8.3.2

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

$y \cdot 1 + y (- 1 x) = \pm e^{C}$

Step 5.8.3.3

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

$y (1 - 1 x) = \pm e^{C}$

Step 5.8.4

Rewrite $- 1 x$ as $- x$ .

$y (1 - x) = \pm e^{C}$

Step 5.8.5

Divide each term in $y (1 - x) = \pm e^{C}$ by $1 - x$ and simplify.

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Step 5.8.5.1

Divide each term in $y (1 - x) = \pm e^{C}$ by $1 - x$ .

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

Step 5.8.5.2

Simplify the left side.

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Step 5.8.5.2.1

Cancel the common factor of $1 - x$ .

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Step 5.8.5.2.1.1

Cancel the common factor.

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

Step 5.8.5.2.1.2

Divide $y$ by $1$ .

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

Step 6

Simplify the constant of integration.

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