Calculus Examples

$\frac{y d y}{t d t} = e^{t^{2}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

$\frac{y}{t} \frac{d y}{d t} = e^{t^{2}}$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Combine $\frac{y}{t}$ and $\frac{d y}{d t}$ .

$\frac{y \frac{d y}{d t}}{t} = e^{t^{2}}$

Step 1.2.2

Multiply both sides by $t$ .

$\frac{y \frac{d y}{d t}}{t} t = e^{t^{2}} t$

Step 1.2.3

Simplify.

Tap for more steps...

Step 1.2.3.1

Simplify the left side.

Tap for more steps...

Step 1.2.3.1.1

Cancel the common factor of $t$ .

Tap for more steps...

Step 1.2.3.1.1.1

Cancel the common factor.

$\frac{y \frac{d y}{d t}}{t} t = e^{t^{2}} t$

Step 1.2.3.1.1.2

Rewrite the expression.

$y \frac{d y}{d t} = e^{t^{2}} t$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Reorder factors in $e^{t^{2}} t$ .

$y \frac{d y}{d t} = t e^{t^{2}}$

Step 1.2.4

Divide each term in $y \frac{d y}{d t} = t e^{t^{2}}$ by $y$ and simplify.

Tap for more steps...

Step 1.2.4.1

Divide each term in $y \frac{d y}{d t} = t e^{t^{2}}$ by $y$ .

$\frac{y \frac{d y}{d t}}{y} = \frac{t e^{t^{2}}}{y}$

Step 1.2.4.2

Simplify the left side.

Tap for more steps...

Step 1.2.4.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{y \frac{d y}{d t}}{y} = \frac{t e^{t^{2}}}{y}$

Step 1.2.4.2.1.2

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

$\frac{d y}{d t} = \frac{t e^{t^{2}}}{y}$

Step 1.3

Multiply both sides by $y$ .

$y \frac{d y}{d t} = y \frac{t e^{t^{2}}}{y}$

Step 1.4

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$y \frac{d y}{d t} = y \frac{t e^{t^{2}}}{y}$

Step 1.4.2

Rewrite the expression.

$y \frac{d y}{d t} = t e^{t^{2}}$

Step 1.5

Rewrite the equation.

$y d y = t e^{t^{2}} d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int y d y = \int t e^{t^{2}} d t$

Step 2.2

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

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

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

Tap for more steps...

Step 2.3.1.1.1

Differentiate $t^{2}$ .

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

Step 2.3.1.1.2

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

$2 t$

Step 2.3.1.2

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

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

Step 2.3.2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 2.3.3

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

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

Step 2.3.4

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 2.3.5

Simplify.

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

Step 2.3.6

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

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

Tap for more steps...

Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (\frac{1}{2} e^{t^{2}} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Combine $\frac{1}{2}$ and $e^{t^{2}}$ .

$y^{2} = 2 (\frac{e^{t^{2}}}{2} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{e^{t^{2}}}{2} + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.3.1

Cancel the common factor.

$y^{2} = 2 \frac{e^{t^{2}}}{2} + 2 K$

Step 3.2.2.1.3.2

Rewrite the expression.

$y^{2} = e^{t^{2}} + 2 K$

Step 3.3

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

$y = \pm \sqrt{e^{t^{2}} + 2 K}$

Step 3.4

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

Tap for more steps...

Step 3.4.1

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

$y = \sqrt{e^{t^{2}} + 2 K}$

Step 3.4.2

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

$y = - \sqrt{e^{t^{2}} + 2 K}$

Step 3.4.3

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

$y = \sqrt{e^{t^{2}} + 2 K}$

$y = - \sqrt{e^{t^{2}} + 2 K}$

$y = \sqrt{e^{t^{2}} + 2 K}$

$y = - \sqrt{e^{t^{2}} + 2 K}$

$y = \sqrt{e^{t^{2}} + 2 K}$

$y = - \sqrt{e^{t^{2}} + 2 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt{e^{t^{2}} + K}$

$y = - \sqrt{e^{t^{2}} + K}$