Calculus Examples

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

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

Apply the distributive property.

$2 x^{2} y \frac{d y}{d x} + 2 x \frac{d y}{d x} + 2 x y^{2} + 2 y = 0$

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 $2 x y^{2}$ from both sides of the equation.

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

Step 1.1.2.2

Subtract $2 y$ from both sides of the equation.

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

Step 1.1.3

Factor $2 x \frac{d y}{d x}$ out of $2 x^{2} y \frac{d y}{d x} + 2 x \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $2 x \frac{d y}{d x}$ out of $2 x^{2} y \frac{d y}{d x}$ .

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

Step 1.1.3.2

Factor $2 x \frac{d y}{d x}$ out of $2 x \frac{d y}{d x}$ .

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

Step 1.1.3.3

Factor $2 x \frac{d y}{d x}$ out of $2 x \frac{d y}{d x} (x y) + 2 x \frac{d y}{d x} \cdot 1$ .

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

Cancel the common factor of $2$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

Rewrite the expression.

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

Step 1.1.4.2.2

Cancel the common factor of $x$ .

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Step 1.1.4.2.2.1

Cancel the common factor.

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

Step 1.1.4.2.2.2

Rewrite the expression.

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

Step 1.1.4.2.3

Cancel the common factor of $x y + 1$ .

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Step 1.1.4.2.3.1

Cancel the common factor.

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

Step 1.1.4.2.3.2

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

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

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Simplify each term.

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Step 1.1.4.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 1.1.4.3.1.1.1

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

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

Step 1.1.4.3.1.1.2

Cancel the common factors.

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Step 1.1.4.3.1.1.2.1

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

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

Step 1.1.4.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.4.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.4.3.1.2

Cancel the common factor of $x$ .

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Step 1.1.4.3.1.2.1

Cancel the common factor.

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

Step 1.1.4.3.1.2.2

Rewrite the expression.

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

Step 1.1.4.3.1.3

Move the negative in front of the fraction.

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

Step 1.1.4.3.1.4

Cancel the common factor of $- 2$ and $2$ .

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Step 1.1.4.3.1.4.1

Factor $2$ out of $- 2 y$ .

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

Step 1.1.4.3.1.4.2

Cancel the common factors.

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Step 1.1.4.3.1.4.2.1

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

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

Step 1.1.4.3.1.4.2.2

Cancel the common factor.

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

Step 1.1.4.3.1.4.2.3

Rewrite the expression.

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

Step 1.1.4.3.1.5

Move the negative in front of the fraction.

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

Step 1.1.4.3.2

To write $- \frac{y^{2}}{x y + 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.1.4.3.3

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

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Step 1.1.4.3.3.1

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

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

Step 1.1.4.3.3.2

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

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

Step 1.1.4.3.4

Combine the numerators over the common denominator.

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

Step 1.1.4.3.5

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

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Step 1.1.4.3.5.1

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

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

Step 1.1.4.3.5.2

Factor $y$ out of $- y$ .

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

Step 1.1.4.3.5.3

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

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

Step 1.1.4.3.6

Cancel the common factor of $- y x - 1$ and $x y + 1$ .

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Step 1.1.4.3.6.1

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

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

Step 1.1.4.3.6.2

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

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

Step 1.1.4.3.6.3

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

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

Step 1.1.4.3.6.4

Rewrite $- (y x + 1)$ as $- 1 (y x + 1)$ .

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

Step 1.1.4.3.6.5

Reorder terms.

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

Step 1.1.4.3.6.6

Cancel the common factor.

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

Step 1.1.4.3.6.7

Rewrite the expression.

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

Step 1.1.4.3.7

Simplify the expression.

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Step 1.1.4.3.7.1

Move $- 1$ to the left of $y$ .

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

Step 1.1.4.3.7.2

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Solve for $y$ .

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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Step 3.6.3.1

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

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

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $x$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

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

Step 3.6.3.2.1.2

Divide $y$ by $1$ .

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

Step 4

Simplify the constant of integration.

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