Calculus Examples

$(5 - x) d y + (y + 3) d x = 0$

Step 1

Subtract $(y + 3) d x$ from both sides of the equation.

$(5 - x) d y = - (y + 3) d x$

Step 2

Multiply both sides by $\frac{1}{(5 - x) (y + 3)}$ .

$\frac{1}{(5 - x) (y + 3)} (5 - x) d y = \frac{1}{(5 - x) (y + 3)} (- (y + 3)) d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Cancel the common factor of $5 - x$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$\frac{1}{(5 - x) (y + 3)} (5 - x) d y = \frac{1}{(5 - x) (y + 3)} (- (y + 3)) d x$

Step 3.1.2

Rewrite the expression.

$\frac{1}{y + 3} d y = \frac{1}{(5 - x) (y + 3)} (- (y + 3)) d x$

Step 3.2

Rewrite using the commutative property of multiplication.

$\frac{1}{y + 3} d y = - \frac{1}{(5 - x) (y + 3)} (y + 3) d x$

Step 3.3

Cancel the common factor of $y + 3$ .

Tap for more steps...

Step 3.3.1

Move the leading negative in $- \frac{1}{(5 - x) (y + 3)}$ into the numerator.

$\frac{1}{y + 3} d y = \frac{- 1}{(5 - x) (y + 3)} (y + 3) d x$

Step 3.3.2

Factor $y + 3$ out of $(5 - x) (y + 3)$ .

$\frac{1}{y + 3} d y = \frac{- 1}{(y + 3) (5 - x)} (y + 3) d x$

Step 3.3.3

Cancel the common factor.

$\frac{1}{y + 3} d y = \frac{- 1}{(y + 3) (5 - x)} (y + 3) d x$

Step 3.3.4

Rewrite the expression.

$\frac{1}{y + 3} d y = \frac{- 1}{5 - x} d x$

Step 3.4

Move the negative in front of the fraction.

$\frac{1}{y + 3} d y = - \frac{1}{5 - x} d x$

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

$\int \frac{1}{y + 3} d y = \int - \frac{1}{5 - x} d x$

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Let $u_{1} = y + 3$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 4.2.1.1

Let $u_{1} = y + 3$ . Find $\frac{d u_{1}}{d y}$ .

Tap for more steps...

Step 4.2.1.1.1

Differentiate $y + 3$ .

$\frac{d}{d y} [y + 3]$

Step 4.2.1.1.2

By the Sum Rule, the derivative of $y + 3$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [3]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [3]$

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 $n = 1$ .

$1 + \frac{d}{d y} [3]$

Step 4.2.1.1.4

Since $3$ is constant with respect to $y$ , the derivative of $3$ 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 $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} d u_{1} = \int - \frac{1}{5 - x} d x$

Step 4.2.2

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

$\ln (| u_{1} |) + C_{1} = \int - \frac{1}{5 - x} d x$

Step 4.2.3

Replace all occurrences of $u_{1}$ with $y + 3$ .

$\ln (| y + 3 |) + C_{1} = \int - \frac{1}{5 - x} d x$

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

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

$\ln (| y + 3 |) + C_{1} = - \int \frac{1}{5 - x} d x$

Step 4.3.2

Let $u_{2} = 5 - x$ . Then $d u_{2} = - d x$ , so $- d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 4.3.2.1

Let $u_{2} = 5 - x$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 4.3.2.1.1

Rewrite.

$\frac{- 1}{1}$

Step 4.3.2.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 4.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\ln (| y + 3 |) + C_{1} = - \int \frac{- 1}{u_{2}} d u_{2}$

Step 4.3.3

Move the negative in front of the fraction.

$\ln (| y + 3 |) + C_{1} = - \int - \frac{1}{u_{2}} d u_{2}$

Step 4.3.4

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\ln (| y + 3 |) + C_{1} = - - \int \frac{1}{u_{2}} d u_{2}$

Step 4.3.5

Simplify.

Tap for more steps...

Step 4.3.5.1

Multiply $- 1$ by $- 1$ .

$\ln (| y + 3 |) + C_{1} = 1 \int \frac{1}{u_{2}} d u_{2}$

Step 4.3.5.2

Multiply $\int \frac{1}{u_{2}} d u_{2}$ by $1$ .

$\ln (| y + 3 |) + C_{1} = \int \frac{1}{u_{2}} d u_{2}$

Step 4.3.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| y + 3 |) + C_{1} = \ln (| u_{2} |) + C_{2}$

Step 4.3.7

Replace all occurrences of $u_{2}$ with $5 - x$ .

$\ln (| y + 3 |) + C_{1} = \ln (| 5 - x |) + C_{2}$

Step 4.4

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

$\ln (| y + 3 |) = \ln (| 5 - x |) + C$

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

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

$\ln (| y + 3 |) - \ln (| 5 - x |) = C$

Step 5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| y + 3 |}{| 5 - x |}) = C$

Step 5.3

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

$e^{\ln (\frac{| y + 3 |}{| 5 - x |})} = e^{C}$

Step 5.4

Rewrite $\ln (\frac{| y + 3 |}{| 5 - x |}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{| y + 3 |}{| 5 - x |}$

Step 5.5

Solve for $y$ .

Tap for more steps...

Step 5.5.1

Rewrite the equation as $\frac{| y + 3 |}{| 5 - x |} = e^{C}$ .

$\frac{| y + 3 |}{| 5 - x |} = e^{C}$

Step 5.5.2

Multiply both sides by $| 5 - x |$ .

$\frac{| y + 3 |}{| 5 - x |} | 5 - x | = e^{C} | 5 - x |$

Step 5.5.3

Simplify the left side.

Tap for more steps...

Step 5.5.3.1

Cancel the common factor of $| 5 - x |$ .

Tap for more steps...

Step 5.5.3.1.1

Cancel the common factor.

$\frac{| y + 3 |}{| 5 - x |} | 5 - x | = e^{C} | 5 - x |$

Step 5.5.3.1.2

Rewrite the expression.

$| y + 3 | = e^{C} | 5 - x |$

Step 5.5.4

Solve for $y$ .

Tap for more steps...

Step 5.5.4.1

Reorder factors in $e^{C} | 5 - x |$ .

$| y + 3 | = | 5 - x | e^{C}$

Step 5.5.4.2

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

$y + 3 = \pm | 5 - x | e^{C}$

Step 5.5.4.3

Reorder factors in $\pm | 5 - x | e^{C}$ .

$y + 3 = \pm e^{C} | 5 - x |$

Step 5.5.4.4

Subtract $3$ from both sides of the equation.

$y = \pm e^{C} | 5 - x | - 3$

Step 6

Group the constant terms together.

Tap for more steps...

Step 6.1

Simplify the constant of integration.

$y = \pm C | 5 - x | - 3$

Step 6.2

Combine constants with the plus or minus.

$y = C | 5 - x | - 3$