Calculus Examples

$\frac{d y}{d t} = - 2 t y + 4 e^{- t^{2}}$ , $y (0) = 3$

Step 1

Add $2 t y$ to both sides of the equation.

$\frac{d y}{d t} + 2 t y = 4 e^{- t^{2}}$

Step 2

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = 2 t$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $2 t$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$e^{2 (\frac{1}{2} t^{2} + C)}$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $2 (\frac{1}{2} t^{2} + C)$ as $2 (\frac{1}{2}) t^{2} + C$ .

$e^{2 (\frac{1}{2}) t^{2} + C}$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$e^{\frac{2}{2} t^{2} + C}$

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

$e^{\frac{2}{2} t^{2} + C}$

Step 2.2.3.2.2.2

Rewrite the expression.

$e^{1 t^{2} + C}$

Step 2.2.3.2.3

Multiply $t^{2}$ by $1$ .

$e^{t^{2} + C}$

Step 2.3

Remove the constant of integration.

$e^{t^{2}}$

Step 3

Multiply each term by the integrating factor $e^{t^{2}}$ .

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Step 3.1

Multiply each term by $e^{t^{2}}$ .

$e^{t^{2}} \frac{d y}{d t} + e^{t^{2}} (2 t y) = e^{t^{2}} (4 e^{- t^{2}})$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = e^{t^{2}} (4 e^{- t^{2}})$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4 e^{t^{2}} e^{- t^{2}}$

Step 3.4

Multiply $e^{t^{2}}$ by $e^{- t^{2}}$ by adding the exponents.

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Step 3.4.1

Move $e^{- t^{2}}$ .

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4 (e^{- t^{2}} e^{t^{2}})$

Step 3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4 e^{- t^{2} + t^{2}}$

Step 3.4.3

Add $- t^{2}$ and $t^{2}$ .

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4 e^{0}$

Step 3.5

Simplify $4 e^{0}$ .

$e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4$

Step 3.6

Reorder factors in $e^{t^{2}} \frac{d y}{d t} + 2 e^{t^{2}} (t y) = 4$ .

$e^{t^{2}} \frac{d y}{d t} + 2 t y e^{t^{2}} = 4$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d t} [e^{t^{2}} y] = 4$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{t^{2}} y] d t = \int 4 d t$

Step 6

Integrate the left side.

$e^{t^{2}} y = \int 4 d t$

Step 7

Apply the constant rule.

$e^{t^{2}} y = 4 t + C$

Step 8

Divide each term in $e^{t^{2}} y = 4 t + C$ by $e^{t^{2}}$ and simplify.

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Step 8.1

Divide each term in $e^{t^{2}} y = 4 t + C$ by $e^{t^{2}}$ .

$\frac{e^{t^{2}} y}{e^{t^{2}}} = \frac{4 t}{e^{t^{2}}} + \frac{C}{e^{t^{2}}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{t^{2}}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{t^{2}} y}{e^{t^{2}}} = \frac{4 t}{e^{t^{2}}} + \frac{C}{e^{t^{2}}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 t}{e^{t^{2}}} + \frac{C}{e^{t^{2}}}$

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $3$ for $y$ in $y = \frac{4 t}{e^{t^{2}}} + \frac{C}{e^{t^{2}}}$ .

$3 = \frac{4 \cdot 0}{e^{0^{2}}} + \frac{C}{e^{0^{2}}}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $\frac{4 \cdot 0}{e^{0^{2}}} + \frac{C}{e^{0^{2}}} = 3$ .

$\frac{4 \cdot 0}{e^{0^{2}}} + \frac{C}{e^{0^{2}}} = 3$

Step 10.2

Simplify $\frac{4 \cdot 0}{e^{0^{2}}} + \frac{C}{e^{0^{2}}}$ .

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Step 10.2.1

Combine fractions.

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Step 10.2.1.1

Combine the numerators over the common denominator.

$\frac{4 \cdot 0 + C}{e^{0^{2}}} = 3$

Step 10.2.1.2

Simplify the expression.

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Step 10.2.1.2.1

Multiply $4$ by $0$ .

$\frac{0 + C}{e^{0^{2}}} = 3$

Step 10.2.1.2.2

Add $0$ and $C$ .

$\frac{C}{e^{0^{2}}} = 3$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{C}{e^{0}} = 3$

Step 10.2.2.2

Anything raised to $0$ is $1$ .

$\frac{C}{1} = 3$

Step 10.2.3

Divide $C$ by $1$ .

$C = 3$

Step 11

Substitute $3$ for $C$ in $y = \frac{4 t}{e^{t^{2}}} + \frac{C}{e^{t^{2}}}$ and simplify.

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Step 11.1

Substitute $3$ for $C$ .

$y = \frac{4 t}{e^{t^{2}}} + \frac{3}{e^{t^{2}}}$