Calculus Examples

$\frac{d y}{d x} = \frac{6 x^{2} y}{e^{2 - x^{3}}}$ , $y (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{6 x^{2}}{e^{2 - x^{3}}} y$

Step 1.2

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (\frac{6 x^{2}}{e^{2 - x^{3}}} y)$

Step 1.3

Cancel the common factor of $y$ .

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Step 1.3.1

Factor $y$ out of $\frac{6 x^{2}}{e^{2 - x^{3}}} y$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{6 x^{2}}{e^{2 - x^{3}}})$

Step 1.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y \frac{6 x^{2}}{e^{2 - x^{3}}})$

Step 1.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = \frac{6 x^{2}}{e^{2 - x^{3}}}$

Step 1.4

Rewrite the equation.

$\frac{1}{y} d y = \frac{6 x^{2}}{e^{2 - x^{3}}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \frac{6 x^{2}}{e^{2 - x^{3}}} d x$

Step 2.2

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

$\ln (| y |) + C_{1} = \int \frac{6 x^{2}}{e^{2 - x^{3}}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\ln (| y |) + C_{1} = 6 \int \frac{x^{2}}{e^{2 - x^{3}}} d x$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Negate the exponent of $e^{2 - x^{3}}$ and move it out of the denominator.

$\ln (| y |) + C_{1} = 6 \int x^{2} {(e^{2 - x^{3}})}^{- 1} d x$

Step 2.3.2.2

Multiply the exponents in ${(e^{2 - x^{3}})}^{- 1}$ .

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Step 2.3.2.2.1

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

$\ln (| y |) + C_{1} = 6 \int x^{2} e^{(2 - x^{3}) \cdot - 1} d x$

Step 2.3.2.2.2

Apply the distributive property.

$\ln (| y |) + C_{1} = 6 \int x^{2} e^{2 \cdot - 1 - x^{3} \cdot - 1} d x$

Step 2.3.2.2.3

Multiply $2$ by $- 1$ .

$\ln (| y |) + C_{1} = 6 \int x^{2} e^{- 2 - x^{3} \cdot - 1} d x$

Step 2.3.2.2.4

Multiply $- x^{3} \cdot - 1$ .

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Step 2.3.2.2.4.1

Multiply $- 1$ by $- 1$ .

$\ln (| y |) + C_{1} = 6 \int x^{2} e^{- 2 + 1 x^{3}} d x$

Step 2.3.2.2.4.2

Multiply $x^{3}$ by $1$ .

$\ln (| y |) + C_{1} = 6 \int x^{2} e^{- 2 + x^{3}} d x$

Step 2.3.3

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

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Step 2.3.3.1

Let $u = - 2 + x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.3.1.1

Differentiate $- 2 + x^{3}$ .

$\frac{d}{d x} [- 2 + x^{3}]$

Step 2.3.3.1.2

By the Sum Rule, the derivative of $- 2 + x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 2] + \frac{d}{d x} [x^{3}]$ .

$\frac{d}{d x} [- 2] + \frac{d}{d x} [x^{3}]$

Step 2.3.3.1.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{3}]$

Step 2.3.3.1.4

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

$0 + 3 x^{2}$

Step 2.3.3.1.5

Add $0$ and $3 x^{2}$ .

$3 x^{2}$

Step 2.3.3.2

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

$\ln (| y |) + C_{1} = 6 \int e^{u} \frac{1}{3} d u$

Step 2.3.4

Combine $e^{u}$ and $\frac{1}{3}$ .

$\ln (| y |) + C_{1} = 6 \int \frac{e^{u}}{3} d u$

Step 2.3.5

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\ln (| y |) + C_{1} = 6 (\frac{1}{3} \int e^{u} d u)$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Combine $\frac{1}{3}$ and $6$ .

$\ln (| y |) + C_{1} = \frac{6}{3} \int e^{u} d u$

Step 2.3.6.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.6.2.1

Factor $3$ out of $6$ .

$\ln (| y |) + C_{1} = \frac{3 \cdot 2}{3} \int e^{u} d u$

Step 2.3.6.2.2

Cancel the common factors.

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Step 2.3.6.2.2.1

Factor $3$ out of $3$ .

$\ln (| y |) + C_{1} = \frac{3 \cdot 2}{3 (1)} \int e^{u} d u$

Step 2.3.6.2.2.2

Cancel the common factor.

$\ln (| y |) + C_{1} = \frac{3 \cdot 2}{3 \cdot 1} \int e^{u} d u$

Step 2.3.6.2.2.3

Rewrite the expression.

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

Step 2.3.6.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.7

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

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

Step 2.3.8

Simplify.

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

Step 2.3.9

Replace all occurrences of $u$ with $- 2 + x^{3}$ .

$\ln (| y |) + C_{1} = 2 e^{- 2 + x^{3}} + C_{2}$

Step 2.4

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

$\ln (| y |) = 2 e^{- 2 + x^{3}} + C$

Step 3

Solve for $y$ .

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Step 3.1

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

$e^{\ln (| y |)} = e^{2 e^{- 2 + x^{3}} + C}$

Step 3.2

Rewrite $\ln (| y |) = 2 e^{- 2 + x^{3}} + C$ 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 $b^{y} = x$ .

$e^{2 e^{- 2 + x^{3}} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{2 e^{- 2 + x^{3}} + C}$ .

$| y | = e^{2 e^{- 2 + x^{3}} + C}$

Step 3.3.2

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

$y = \pm e^{2 e^{- 2 + x^{3}} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{2 e^{- 2 + x^{3}} + C}$ as $e^{2 e^{- 2 + x^{3}}} e^{C}$ .

$y = \pm e^{2 e^{- 2 + x^{3}}} e^{C}$

Step 4.2

Reorder $e^{2 e^{- 2 + x^{3}}}$ and $e^{C}$ .

$y = \pm e^{C} e^{2 e^{- 2 + x^{3}}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{2 e^{- 2 + x^{3}}}$

Step 5

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $1$ for $y$ in $y = C e^{2 e^{- 2 + x^{3}}}$ .

$1 = C e^{2 e^{- 2 + 0^{3}}}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{2 e^{- 2 + 0^{3}}} = 1$ .

$C e^{2 e^{- 2 + 0^{3}}} = 1$

Step 6.2

Simplify.

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Step 6.2.1

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

$C e^{2 e^{- 2 + 0}} = 1$

Step 6.2.2

Add $- 2$ and $0$ .

$C e^{2 e^{- 2}} = 1$

Step 6.2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$C e^{2 \frac{1}{e^{2}}} = 1$

Step 6.2.4

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

$C e^{\frac{2}{e^{2}}} = 1$

Step 6.3

Divide each term in $C e^{\frac{2}{e^{2}}} = 1$ by $e^{\frac{2}{e^{2}}}$ and simplify.

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Step 6.3.1

Divide each term in $C e^{\frac{2}{e^{2}}} = 1$ by $e^{\frac{2}{e^{2}}}$ .

$\frac{C e^{\frac{2}{e^{2}}}}{e^{\frac{2}{e^{2}}}} = \frac{1}{e^{\frac{2}{e^{2}}}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor.

$\frac{C e^{\frac{2}{e^{2}}}}{e^{\frac{2}{e^{2}}}} = \frac{1}{e^{\frac{2}{e^{2}}}}$

Step 6.3.2.2

Divide $C$ by $1$ .

$C = \frac{1}{e^{\frac{2}{e^{2}}}}$

Step 7

Substitute $\frac{1}{e^{\frac{2}{e^{2}}}}$ for $C$ in $y = C e^{2 e^{- 2 + x^{3}}}$ and simplify.

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Step 7.1

Substitute $\frac{1}{e^{\frac{2}{e^{2}}}}$ for $C$ .

$y = \frac{1}{e^{\frac{2}{e^{2}}}} e^{2 e^{- 2 + x^{3}}}$

Step 7.2

Combine $\frac{1}{e^{\frac{2}{e^{2}}}}$ and $e^{2 e^{- 2 + x^{3}}}$ .

$y = \frac{e^{2 e^{- 2 + x^{3}}}}{e^{\frac{2}{e^{2}}}}$

Step 7.3

Factor $e^{\frac{2}{e^{2}}}$ out of $e^{2 e^{- 2 + x^{3}}}$ .

$y = \frac{e^{\frac{2}{e^{2}}} e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}}{e^{\frac{2}{e^{2}}}}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Multiply by $1$ .

$y = \frac{e^{\frac{2}{e^{2}}} e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}}{e^{\frac{2}{e^{2}}} \cdot 1}$

Step 7.4.2

Cancel the common factor.

$y = \frac{e^{\frac{2}{e^{2}}} e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}}{e^{\frac{2}{e^{2}}} \cdot 1}$

Step 7.4.3

Rewrite the expression.

$y = \frac{e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}}{1}$

Step 7.4.4

Divide $e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}$ by $1$ .

$y = e^{\frac{2 e^{- 2 + x^{3}} e^{2} - 2}{e^{2}}}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Factor $2$ out of $2 e^{- 2 + x^{3}} e^{2} - 2$ .

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Step 7.5.1.1

Factor $2$ out of $2 e^{- 2 + x^{3}} e^{2}$ .

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

Step 7.5.1.2

Factor $2$ out of $- 2$ .

$y = e^{\frac{2 (e^{- 2 + x^{3}} e^{2}) + 2 (- 1)}{e^{2}}}$

Step 7.5.1.3

Factor $2$ out of $2 (e^{- 2 + x^{3}} e^{2}) + 2 (- 1)$ .

$y = e^{\frac{2 (e^{- 2 + x^{3}} e^{2} - 1)}{e^{2}}}$

Step 7.5.2

Multiply $e^{- 2 + x^{3}}$ by $e^{2}$ by adding the exponents.

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Step 7.5.2.1

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

$y = e^{\frac{2 (e^{- 2 + x^{3} + 2} - 1)}{e^{2}}}$

Step 7.5.2.2

Combine the opposite terms in $- 2 + x^{3} + 2$ .

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Step 7.5.2.2.1

Add $- 2$ and $2$ .

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

Step 7.5.2.2.2

Add $x^{3}$ and $0$ .

$y = e^{\frac{2 (e^{x^{3}} - 1)}{e^{2}}}$