Calculus Examples

$(y + 4) d x + x d y = 0$

Step 1

Subtract $(y + 4) d x$ from both sides of the equation.

$x d y = - (y + 4) d x$

Step 2

Multiply both sides by $\frac{1}{x (y + 4)}$ .

$\frac{1}{x (y + 4)} x d y = \frac{1}{x (y + 4)} (- (y + 4)) d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

$\frac{1}{x (y + 4)} x d y = \frac{1}{x (y + 4)} (- (y + 4)) d x$

Step 3.1.2

Rewrite the expression.

$\frac{1}{y + 4} d y = \frac{1}{x (y + 4)} (- (y + 4)) d x$

Step 3.2

Rewrite using the commutative property of multiplication.

$\frac{1}{y + 4} d y = - \frac{1}{x (y + 4)} (y + 4) d x$

Step 3.3

Cancel the common factor of $y + 4$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{x (y + 4)}$ into the numerator.

$\frac{1}{y + 4} d y = \frac{- 1}{x (y + 4)} (y + 4) d x$

Step 3.3.2

Factor $y + 4$ out of $x (y + 4)$ .

$\frac{1}{y + 4} d y = \frac{- 1}{(y + 4) x} (y + 4) d x$

Step 3.3.3

Cancel the common factor.

$\frac{1}{y + 4} d y = \frac{- 1}{(y + 4) x} (y + 4) d x$

Step 3.3.4

Rewrite the expression.

$\frac{1}{y + 4} d y = \frac{- 1}{x} d x$

Step 3.4

Move the negative in front of the fraction.

$\frac{1}{y + 4} d y = - \frac{1}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{1}{y + 4} d y = \int - \frac{1}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Let $u = y + 4$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.1.1

Let $u = y + 4$ . Find $\frac{d u}{d y}$ .

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Step 4.2.1.1.1

Differentiate $y + 4$ .

$\frac{d}{d y} [y + 4]$

Step 4.2.1.1.2

By the Sum Rule, the derivative of $y + 4$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [4]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [4]$

Step 4.2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [4]$

Step 4.2.1.1.4

Since $4$ is constant with respect to $y$ , the derivative of $4$ with respect to $y$ is $0$ .

$1 + 0$

Step 4.2.1.1.5

Add $1$ and $0$ .

$1$

Step 4.2.1.2

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

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

Step 4.2.2

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

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

Step 4.2.3

Replace all occurrences of $u$ with $y + 4$ .

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

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.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

$\ln (| y + 4 |) = - \ln (| x |) + C$

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.

$\ln (| y + 4 |) + \ln (| x |) = C$

Step 5.2

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

$\ln (| y + 4 | | x |) = C$

Step 5.3

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

$\ln (| (y + 4) x |) = C$

Step 5.4

Apply the distributive property.

$\ln (| y x + 4 x |) = C$

Step 5.5

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

$e^{\ln (| y x + 4 x |)} = e^{C}$

Step 5.6

Rewrite $\ln (| y x + 4 x |) = C$ 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 $b^{y} = x$ .

$e^{C} = | y x + 4 x |$

Step 5.7

Solve for $y$ .

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Step 5.7.1

Rewrite the equation as $| y x + 4 x | = e^{C}$ .

$| y x + 4 x | = e^{C}$

Step 5.7.2

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

$y x + 4 x = \pm e^{C}$

Step 5.7.3

Subtract $4 x$ from both sides of the equation.

$y x = \pm e^{C} - 4 x$

Step 5.7.4

Divide each term in $y x = \pm e^{C} - 4 x$ by $x$ and simplify.

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Step 5.7.4.1

Divide each term in $y x = \pm e^{C} - 4 x$ by $x$ .

$\frac{y x}{x} = \frac{\pm e^{C}}{x} + \frac{- 4 x}{x}$

Step 5.7.4.2

Simplify the left side.

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Step 5.7.4.2.1

Cancel the common factor of $x$ .

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Step 5.7.4.2.1.1

Cancel the common factor.

$\frac{y x}{x} = \frac{\pm e^{C}}{x} + \frac{- 4 x}{x}$

Step 5.7.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\pm e^{C}}{x} + \frac{- 4 x}{x}$

Step 5.7.4.3

Simplify the right side.

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Step 5.7.4.3.1

Cancel the common factor of $x$ .

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Step 5.7.4.3.1.1

Cancel the common factor.

$y = \frac{\pm e^{C}}{x} + \frac{- 4 x}{x}$

Step 5.7.4.3.1.2

Divide $- 4$ by $1$ .

$y = \frac{\pm e^{C}}{x} - 4$

Step 6

Simplify the constant of integration.

$y = \frac{C}{x} - 4$