Calculus Examples

$\frac{2 d y}{d x} + y = 3$

Step 1

Separate the variables.

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Step 1.1

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

$2 \frac{d y}{d x} + y = 3$

Step 1.2

Solve for $\frac{d y}{d x}$ .

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Step 1.2.1

Subtract $y$ from both sides of the equation.

$2 \frac{d y}{d x} = 3 - y$

Step 1.2.2

Divide each term in $2 \frac{d y}{d x} = 3 - y$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 \frac{d y}{d x} = 3 - y$ by $2$ .

$\frac{2 \frac{d y}{d x}}{2} = \frac{3}{2} + \frac{- y}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 \frac{d y}{d x}}{2} = \frac{3}{2} + \frac{- y}{2}$

Step 1.2.2.2.1.2

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

$\frac{d y}{d x} = \frac{3}{2} + \frac{- y}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{3}{2} - \frac{y}{2}$

Step 1.3

Multiply both sides by $\frac{1}{\frac{3}{2} - \frac{y}{2}}$ .

$\frac{1}{\frac{3}{2} - \frac{y}{2}} \frac{d y}{d x} = \frac{1}{\frac{3}{2} - \frac{y}{2}} (\frac{3}{2} - \frac{y}{2})$

Step 1.4

Cancel the common factor of $\frac{3}{2} - \frac{y}{2}$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{\frac{3}{2} - \frac{y}{2}} \frac{d y}{d x} = \frac{1}{\frac{3}{2} - \frac{y}{2}} (\frac{3}{2} - \frac{y}{2})$

Step 1.4.2

Rewrite the expression.

$\frac{1}{\frac{3}{2} - \frac{y}{2}} \frac{d y}{d x} = 1$

Step 1.5

Rewrite the equation.

$\frac{1}{\frac{3}{2} - \frac{y}{2}} d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\frac{3}{2} - \frac{y}{2}} d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = \frac{3}{2} - \frac{y}{2}$ . Then $d u = - \frac{1}{2} d y$ , so $- 2 d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = \frac{3}{2} - \frac{y}{2}$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $\frac{3}{2} - \frac{y}{2}$ .

$\frac{d}{d y} [\frac{3}{2} - \frac{y}{2}]$

Step 2.2.1.1.2

Differentiate.

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Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $\frac{3}{2} - \frac{y}{2}$ with respect to $y$ is $\frac{d}{d y} [\frac{3}{2}] + \frac{d}{d y} [- \frac{y}{2}]$ .

$\frac{d}{d y} [\frac{3}{2}] + \frac{d}{d y} [- \frac{y}{2}]$

Step 2.2.1.1.2.2

Since $\frac{3}{2}$ is constant with respect to $y$ , the derivative of $\frac{3}{2}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [- \frac{y}{2}]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [- \frac{y}{2}]$ .

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Step 2.2.1.1.3.1

Since $- \frac{1}{2}$ is constant with respect to $y$ , the derivative of $- \frac{y}{2}$ with respect to $y$ is $- \frac{1}{2} \frac{d}{d y} [y]$ .

$0 - \frac{1}{2} \frac{d}{d y} [y]$

Step 2.2.1.1.3.2

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

$0 - \frac{1}{2} \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 1$ by $1$ .

$0 - \frac{1}{2}$

Step 2.2.1.1.4

Subtract $\frac{1}{2}$ from $0$ .

$- \frac{1}{2}$

Step 2.2.1.2

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

$\int \frac{- 1}{u} \cdot \frac{- 1}{- \frac{1}{2}} d u = \int d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

$\int - \frac{1}{u} \cdot \frac{- 1}{- \frac{1}{2}} d u = \int d x$

Step 2.2.2.2

Dividing two negative values results in a positive value.

$\int - \frac{1}{u} \cdot \frac{1}{\frac{1}{2}} d u = \int d x$

Step 2.2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int - \frac{1}{u} (1 \cdot 2) d u = \int d x$

Step 2.2.2.4

Multiply $2$ by $1$ .

$\int - \frac{1}{u} \cdot 2 d u = \int d x$

Step 2.2.2.5

Multiply $2$ by $- 1$ .

$\int - 2 \frac{1}{u} d u = \int d x$

Step 2.2.2.6

Combine $- 2$ and $\frac{1}{u}$ .

$\int \frac{- 2}{u} d u = \int d x$

Step 2.2.2.7

Move the negative in front of the fraction.

$\int - \frac{2}{u} d u = \int d x$

Step 2.2.3

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

$- \int \frac{2}{u} d u = \int d x$

Step 2.2.4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$- (2 \int \frac{1}{u} d u) = \int d x$

Step 2.2.5

Multiply $2$ by $- 1$ .

$- 2 \int \frac{1}{u} d u = \int d x$

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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

Step 2.2.8

Replace all occurrences of $u$ with $\frac{3}{2} - \frac{y}{2}$ .

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

Step 2.2.9

Reorder terms.

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- 2 \ln (| \frac{3}{2} - \frac{1}{2} y |) = x + C$ by $- 2$ and simplify.

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Step 3.1.1

Divide each term in $- 2 \ln (| \frac{3}{2} - \frac{1}{2} y |) = x + C$ by $- 2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

Divide $\ln (| \frac{3}{2} - \frac{1}{2} y |)$ by $1$ .

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

Step 3.1.2.2

Combine $y$ and $\frac{1}{2}$ .

$\ln (| \frac{3}{2} - \frac{y}{2} |) = \frac{x}{- 2} + \frac{C}{- 2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| \frac{3}{2} - \frac{y}{2} |) = - \frac{x}{2} + \frac{C}{- 2}$

Step 3.1.3.1.2

Move the negative in front of the fraction.

$\ln (| \frac{3}{2} - \frac{y}{2} |) = - \frac{x}{2} - \frac{C}{2}$

Step 3.2

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

$e^{\ln (| \frac{3}{2} - \frac{y}{2} |)} = e^{- \frac{x}{2} - \frac{C}{2}}$

Step 3.3

Rewrite $\ln (| \frac{3}{2} - \frac{y}{2} |) = - \frac{x}{2} - \frac{C}{2}$ 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 $b^{y} = x$ .

$e^{- \frac{x}{2} - \frac{C}{2}} = | \frac{3}{2} - \frac{y}{2} |$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $| \frac{3}{2} - \frac{y}{2} | = e^{- \frac{x}{2} - \frac{C}{2}}$ .

$| \frac{3}{2} - \frac{y}{2} | = e^{- \frac{x}{2} - \frac{C}{2}}$

Step 3.4.2

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

$\frac{3}{2} - \frac{y}{2} = \pm e^{- \frac{x}{2} - \frac{C}{2}}$

Step 3.4.3

Subtract $\frac{3}{2}$ from both sides of the equation.

$- \frac{y}{2} = \pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2}$

Step 3.4.4

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{y}{2}) = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5

Simplify both sides of the equation.

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Step 3.4.5.1

Simplify the left side.

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Step 3.4.5.1.1

Simplify $- 2 (- \frac{y}{2})$ .

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Step 3.4.5.1.1.1

Cancel the common factor of $2$ .

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Step 3.4.5.1.1.1.1

Move the leading negative in $- \frac{y}{2}$ into the numerator.

$- 2 \frac{- y}{2} = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- y}{2} = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- y}{2} = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.1.1.1.4

Rewrite the expression.

$- - y = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.1.1.2

Multiply.

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Step 3.4.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.1.1.2.2

Multiply $y$ by $1$ .

$y = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$

Step 3.4.5.2

Simplify the right side.

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Step 3.4.5.2.1

Simplify $- 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}} - \frac{3}{2})$ .

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Step 3.4.5.2.1.1

To write $\pm e^{- \frac{x}{2} - \frac{C}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - 2 ((\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot \frac{2}{2} - \frac{3}{2})$

Step 3.4.5.2.1.2

Simplify terms.

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Step 3.4.5.2.1.2.1

Combine $\pm e^{- \frac{x}{2} - \frac{C}{2}}$ and $\frac{2}{2}$ .

$y = - 2 (\frac{(\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot 2}{2} - \frac{3}{2})$

Step 3.4.5.2.1.2.2

Combine the numerators over the common denominator.

$y = - 2 \frac{(\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot 2 - 3}{2}$

Step 3.4.5.2.1.2.3

Cancel the common factor of $2$ .

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Step 3.4.5.2.1.2.3.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \frac{(\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot 2 - 3}{2}$

Step 3.4.5.2.1.2.3.2

Cancel the common factor.

$y = 2 \cdot - 1 \frac{(\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot 2 - 3}{2}$

Step 3.4.5.2.1.2.3.3

Rewrite the expression.

$y = - ((\pm e^{- \frac{x}{2} - \frac{C}{2}}) \cdot 2 - 3)$

Step 3.4.5.2.1.3

Move $2$ to the left of $\pm e^{- \frac{x}{2} - \frac{C}{2}}$ .

$y = - (2 (\pm e^{- \frac{x}{2} - \frac{C}{2}}) - 3)$

Step 3.4.5.2.1.4

Simplify by multiplying through.

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Step 3.4.5.2.1.4.1

Apply the distributive property.

$y = - (2 (\pm e^{- \frac{x}{2} - \frac{C}{2}})) - - 3$

Step 3.4.5.2.1.4.2

Multiply.

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Step 3.4.5.2.1.4.2.1

Multiply $2$ by $- 1$ .

$y = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}}) - - 3$

Step 3.4.5.2.1.4.2.2

Multiply $- 1$ by $- 3$ .

$y = - 2 (\pm e^{- \frac{x}{2} - \frac{C}{2}}) + 3$

Step 3.4.6

Reorder $- \frac{x}{2}$ and $- \frac{C}{2}$ .

$y = - 2 (\pm e^{- \frac{C}{2} - \frac{x}{2}}) + 3$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = - 2 (\pm e^{C - \frac{x}{2}}) + 3$

Step 4.2

Reorder terms.

$y = - 2 (\pm e^{- \frac{x}{2} + C}) + 3$

Step 4.3

Rewrite $e^{- \frac{x}{2} + C}$ as $e^{- \frac{x}{2}} e^{C}$ .

$y = - 2 (\pm e^{- \frac{x}{2}} e^{C}) + 3$

Step 4.4

Reorder $e^{- \frac{x}{2}}$ and $e^{C}$ .

$y = - 2 (\pm e^{C} e^{- \frac{x}{2}}) + 3$

Step 4.5

Combine constants with the plus or minus.

$y = - 2 (C e^{- \frac{x}{2}}) + 3$