Calculus Examples

$t \frac{d x}{d t} - x = 3$

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Add $x$ to both sides of the equation.

$t \frac{d x}{d t} = 3 + x$

Step 1.1.2

Divide each term in $t \frac{d x}{d t} = 3 + x$ by $t$ and simplify.

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Step 1.1.2.1

Divide each term in $t \frac{d x}{d t} = 3 + x$ by $t$ .

$\frac{t \frac{d x}{d t}}{t} = \frac{3}{t} + \frac{x}{t}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $t$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{t \frac{d x}{d t}}{t} = \frac{3}{t} + \frac{x}{t}$

Step 1.1.2.2.1.2

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

$\frac{d x}{d t} = \frac{3}{t} + \frac{x}{t}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{d x}{d t} = \frac{3 + x}{t}$

Step 1.3

Multiply both sides by $\frac{1}{3 + x}$ .

$\frac{1}{3 + x} \frac{d x}{d t} = \frac{1}{3 + x} \cdot \frac{3 + x}{t}$

Step 1.4

Cancel the common factor of $3 + x$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1}{3 + x} \frac{d x}{d t} = \frac{1}{3 + x} \cdot \frac{3 + x}{t}$

Step 1.4.2

Rewrite the expression.

$\frac{1}{3 + x} \frac{d x}{d t} = \frac{1}{t}$

Step 1.5

Rewrite the equation.

$\frac{1}{3 + x} d x = \frac{1}{t} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{3 + x} d x = \int \frac{1}{t} d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 3 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 3 + x$ . Find $\frac{d u}{d x}$ .

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Step 2.2.1.1.1

Differentiate $3 + x$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 2.2.1.1.4

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

$0 + 1$

Step 2.2.1.1.5

Add $0$ and $1$ .

$1$

Step 2.2.1.2

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

$\int \frac{1}{u} d u = \int \frac{1}{t} d t$

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $3 + x$ .

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

Step 2.3

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

$\ln (| 3 + x |) + C_{1} = \ln (| t |) + C_{2}$

Step 2.4

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

$\ln (| 3 + x |) = \ln (| t |) + C$

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

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

$\ln (\frac{| 3 + x |}{| t |}) = C$

Step 3.3

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

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

Step 3.4

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

$e^{C} = \frac{| 3 + x |}{| t |}$

Step 3.5

Solve for $x$ .

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Step 3.5.1

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

$\frac{| 3 + x |}{| t |} = e^{C}$

Step 3.5.2

Multiply both sides by $| t |$ .

$\frac{| 3 + x |}{| t |} | t | = e^{C} | t |$

Step 3.5.3

Simplify the left side.

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Step 3.5.3.1

Cancel the common factor of $| t |$ .

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Step 3.5.3.1.1

Cancel the common factor.

$\frac{| 3 + x |}{| t |} | t | = e^{C} | t |$

Step 3.5.3.1.2

Rewrite the expression.

$| 3 + x | = e^{C} | t |$

Step 3.5.4

Solve for $x$ .

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Step 3.5.4.1

Reorder factors in $e^{C} | t |$ .

$| 3 + x | = | t | e^{C}$

Step 3.5.4.2

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

$3 + x = \pm | t | e^{C}$

Step 3.5.4.3

Reorder factors in $\pm | t | e^{C}$ .

$3 + x = \pm e^{C} | t |$

Step 3.5.4.4

Subtract $3$ from both sides of the equation.

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

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$x = \pm C | t | - 3$

Step 4.2

Combine constants with the plus or minus.

$x = C | t | - 3$