Calculus Examples

$\frac{d y}{d x} + \frac{5 x^{2}}{e^{3 x^{3}}} y = 0$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{5 x^{2}}{e^{3 x^{3}}}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{5 x^{2}}{e^{3 x^{3}}} d x}$

Step 1.2

Integrate $\frac{5 x^{2}}{e^{3 x^{3}}}$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{5 \int \frac{x^{2}}{e^{3 x^{3}}} d x}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Negate the exponent of $e^{3 x^{3}}$ and move it out of the denominator.

$e^{5 \int x^{2} {(e^{3 x^{3}})}^{- 1} d x}$

Step 1.2.2.2

Multiply the exponents in ${(e^{3 x^{3}})}^{- 1}$ .

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Step 1.2.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{5 \int x^{2} e^{3 x^{3} \cdot - 1} d x}$

Step 1.2.2.2.2

Multiply $- 1$ by $3$ .

$e^{5 \int x^{2} e^{- 3 x^{3}} d x}$

Step 1.2.3

Let $u_{2} = e^{- 3 x^{3}}$ . Then $d u_{2} = - 9 x^{2} e^{- 3 x^{3}} d x$ , so $- \frac{1}{9} d u_{2} = x^{2} e^{- 3 x^{3}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.2.3.1

Let $u_{2} = e^{- 3 x^{3}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.2.3.1.1

Differentiate $e^{- 3 x^{3}}$ .

$\frac{d}{d x} [e^{- 3 x^{3}}]$

Step 1.2.3.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 3 x^{3}$ .

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Step 1.2.3.1.2.1

To apply the Chain Rule, set $u_{1}$ as $- 3 x^{3}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- 3 x^{3}]$

Step 1.2.3.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- 3 x^{3}]$

Step 1.2.3.1.2.3

Replace all occurrences of $u_{1}$ with $- 3 x^{3}$ .

$e^{- 3 x^{3}} \frac{d}{d x} [- 3 x^{3}]$

Step 1.2.3.1.3

Differentiate.

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Step 1.2.3.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{3}]$ .

$e^{- 3 x^{3}} (- 3 \frac{d}{d x} [x^{3}])$

Step 1.2.3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$e^{- 3 x^{3}} (- 3 (3 x^{2}))$

Step 1.2.3.1.3.3

Multiply $3$ by $- 3$ .

$e^{- 3 x^{3}} (- 9 x^{2})$

Step 1.2.3.1.4

Simplify.

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Step 1.2.3.1.4.1

Reorder the factors of $e^{- 3 x^{3}} (- 9 x^{2})$ .

$- 9 e^{- 3 x^{3}} x^{2}$

Step 1.2.3.1.4.2

Reorder factors in $- 9 e^{- 3 x^{3}} x^{2}$ .

$- 9 x^{2} e^{- 3 x^{3}}$

Step 1.2.3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$e^{5 \int \frac{1}{- 9} d u_{2}}$

Step 1.2.4

Move the negative in front of the fraction.

$e^{5 \int - \frac{1}{9} d u_{2}}$

Step 1.2.5

Apply the constant rule.

$e^{5 (- \frac{1}{9} u_{2} + C)}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify.

$e^{5 (- \frac{1}{9} u_{2}) + C}$

Step 1.2.6.2

Simplify.

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Step 1.2.6.2.1

Combine $u_{2}$ and $\frac{1}{9}$ .

$e^{5 (- \frac{u_{2}}{9}) + C}$

Step 1.2.6.2.2

Multiply $- 1$ by $5$ .

$e^{- 5 \frac{u_{2}}{9} + C}$

Step 1.2.6.2.3

Combine $- 5$ and $\frac{u_{2}}{9}$ .

$e^{\frac{- 5 u_{2}}{9} + C}$

Step 1.2.6.2.4

Move the negative in front of the fraction.

$e^{- \frac{5 u_{2}}{9} + C}$

Step 1.2.6.3

Replace all occurrences of $u_{2}$ with $e^{- 3 x^{3}}$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9} + C}$

Step 1.2.6.4

Reorder terms.

$e^{- \frac{5}{9} e^{- 3 x^{3}} + C}$

Step 1.3

Remove the constant of integration.

$e^{- \frac{5}{9} e^{- 3 x^{3}}}$

Step 1.4

Combine $e^{- 3 x^{3}}$ and $\frac{5}{9}$ .

$e^{- \frac{e^{- 3 x^{3}} \cdot 5}{9}}$

Step 1.5

Move $5$ to the left of $e^{- 3 x^{3}}$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}}$

Step 2

Multiply each term by the integrating factor $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ .

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Step 2.1

Multiply each term by $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + e^{- \frac{5 e^{- 3 x^{3}}}{9}} (\frac{5 x^{2}}{e^{3 x^{3}}} y) = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{5 x^{2}}{e^{3 x^{3}}}$ and $y$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{5 x^{2} y}{e^{3 x^{3}}} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.2

Combine $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ and $\frac{5 x^{2} y}{e^{3 x^{3}}}$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + \frac{e^{- \frac{5 e^{- 3 x^{3}}}{9}} (5 x^{2} y)}{e^{3 x^{3}}} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.3

Cancel the common factor of $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ and $e^{3 x^{3}}$ .

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Step 2.2.3.1

Factor $e^{3 x^{3}}$ out of $e^{- \frac{5 e^{- 3 x^{3}}}{9}} (5 x^{2} y)$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + \frac{e^{3 x^{3}} (e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y))}{e^{3 x^{3}}} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Multiply by $1$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + \frac{e^{3 x^{3}} (e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y))}{e^{3 x^{3}} \cdot 1} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.3.2.2

Cancel the common factor.

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + \frac{e^{3 x^{3}} (e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y))}{e^{3 x^{3}} \cdot 1} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.3.2.3

Rewrite the expression.

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + \frac{e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y)}{1} = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.3.2.4

Divide $e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y)$ by $1$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (5 x^{2} y) = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.2.4

Rewrite using the commutative property of multiplication.

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + 5 e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (x^{2} y) = e^{- \frac{5 e^{- 3 x^{3}}}{9}} \cdot 0$

Step 2.3

Multiply $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ by $0$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + 5 e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (x^{2} y) = 0$

Step 2.4

Reorder factors in $e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + 5 e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} (x^{2} y) = 0$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} \frac{d y}{d x} + 5 x^{2} y e^{- \frac{5 e^{- 3 x^{3}}}{9} - 3 x^{3}} = 0$

Step 3

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

$\frac{d}{d x} [e^{- \frac{5 e^{- 3 x^{3}}}{9}} y] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- \frac{5 e^{- 3 x^{3}}}{9}} y] d x = \int 0 d x$

Step 5

Integrate the left side.

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} y = \int 0 d x$

Step 6

Integrate the right side.

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Step 6.1

The integral of $0$ with respect to $x$ is $0$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} y = 0 + C$

Step 6.2

Add $0$ and $C$ .

$e^{- \frac{5 e^{- 3 x^{3}}}{9}} y = C$

Step 7

Divide each term in $e^{- \frac{5 e^{- 3 x^{3}}}{9}} y = C$ by $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ and simplify.

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Step 7.1

Divide each term in $e^{- \frac{5 e^{- 3 x^{3}}}{9}} y = C$ by $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ .

$\frac{e^{- \frac{5 e^{- 3 x^{3}}}{9}} y}{e^{- \frac{5 e^{- 3 x^{3}}}{9}}} = \frac{C}{e^{- \frac{5 e^{- 3 x^{3}}}{9}}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{- \frac{5 e^{- 3 x^{3}}}{9}}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{- \frac{5 e^{- 3 x^{3}}}{9}} y}{e^{- \frac{5 e^{- 3 x^{3}}}{9}}} = \frac{C}{e^{- \frac{5 e^{- 3 x^{3}}}{9}}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{C}{e^{- \frac{5 e^{- 3 x^{3}}}{9}}}$