Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d t} = \frac{2 t}{1 + t^{2}} \cdot \frac{1}{y}$

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Factor $y$ out of $(1 + t^{2}) y$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

$y \frac{d y}{d t} = \frac{2 t}{1 + t^{2}}$

Step 1.4

Rewrite the equation.

$y d y = \frac{2 t}{1 + t^{2}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{2 t}{1 + t^{2}} d t$

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{2 t}{1 + t^{2}} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = 2 \int \frac{t}{1 + t^{2}} d t$

Step 2.3.2

Let $u = 1 + t^{2}$ . Then $d u = 2 t d t$ , so $\frac{1}{2} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 1 + t^{2}$ . Find $\frac{d u}{d t}$ .

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Step 2.3.2.1.1

Differentiate $1 + t^{2}$ .

$\frac{d}{d t} [1 + t^{2}]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $1 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [t^{2}]$ .

$\frac{d}{d t} [1] + \frac{d}{d t} [t^{2}]$

Step 2.3.2.1.3

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [t^{2}]$

Step 2.3.2.1.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$0 + 2 t$

Step 2.3.2.1.5

Add $0$ and $2 t$ .

$2 t$

Step 2.3.2.2

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

$\frac{1}{2} y^{2} + C_{1} = 2 \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = 2 \int \frac{1}{u \cdot 2} d u$

Step 2.3.3.2

Move $2$ to the left of $u$ .

$\frac{1}{2} y^{2} + C_{1} = 2 \int \frac{1}{2 u} d u$

Step 2.3.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = 2 (\frac{1}{2} \int \frac{1}{u} d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{2} \int \frac{1}{u} d u$

Step 2.3.5.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.1

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{2} \int \frac{1}{u} d u$

Step 2.3.5.2.2

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = 1 \int \frac{1}{u} d u$

Step 2.3.5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{u} d u$

Step 2.3.6

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

$\frac{1}{2} y^{2} + C_{1} = \ln (| u |) + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $1 + t^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \ln (| 1 + t^{2} |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \ln (| 1 + t^{2} |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\ln (| 1 + t^{2} |) + C)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (\ln (| 1 + t^{2} |) + C)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\ln (| 1 + t^{2} |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\ln (| 1 + t^{2} |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{2} = 2 \ln (| 1 + t^{2} |) + 2 C$

Step 3.3

Simplify $2 \ln (| 1 + t^{2} |)$ by moving $2$ inside the logarithm.

$y^{2} = \ln ({| 1 + t^{2} |}^{2}) + 2 C$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\ln ({| 1 + t^{2} |}^{2}) + 2 C}$

Step 3.5

Remove the absolute value in ${| 1 + t^{2} |}^{2}$ because exponentiations with even powers are always positive.

$y = \pm \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

$y = \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

$y = \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

$y = - \sqrt{\ln ({(1 + t^{2})}^{2}) + 2 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt{\ln ({(1 + t^{2})}^{2}) + C}$

$y = - \sqrt{\ln ({(1 + t^{2})}^{2}) + C}$