Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Subtract $2 x y^{3}$ from both sides of the equation.

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

Step 1.1.2

Divide each term in $y^{2} \frac{d y}{d x} = 6 x - 2 x y^{3}$ by $y^{2}$ and simplify.

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Step 1.1.2.1

Divide each term in $y^{2} \frac{d y}{d x} = 6 x - 2 x y^{3}$ by $y^{2}$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $y^{2}$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

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

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 1.1.2.3.1.1

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

$\frac{d y}{d x} = \frac{6 x}{y^{2}} + \frac{y^{2} (- 2 x y)}{y^{2}}$

Step 1.1.2.3.1.2

Cancel the common factors.

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Step 1.1.2.3.1.2.1

Multiply by $1$ .

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

Step 1.1.2.3.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.2.4

Divide $- 2 x y$ by $1$ .

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

Step 1.2

Factor.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Factor $2 x$ out of $- 2 x y$ .

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

Step 1.2.1.3

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

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

Step 1.2.2

Rewrite $- 1 y$ as $- y$ .

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

Step 1.2.3

To write $- y$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

$\frac{d y}{d x} = 2 x (\frac{3}{y^{2}} - y \cdot \frac{y^{2}}{y^{2}})$

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 1.2.6.1

Move $y^{2}$ .

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

Step 1.2.6.2

Multiply $y^{2}$ by $y$ .

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Step 1.2.6.2.1

Raise $y$ to the power of $1$ .

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

Step 1.2.6.2.2

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

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

Step 1.2.6.3

Add $2$ and $1$ .

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

Step 1.2.7

Combine exponents.

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Step 1.2.7.1

Combine $2$ and $\frac{3 - y^{3}}{y^{2}}$ .

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

Step 1.2.7.2

Combine $x$ and $\frac{2 (3 - y^{3})}{y^{2}}$ .

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

Step 1.2.8

Remove unnecessary parentheses.

$\frac{d y}{d x} = \frac{x \cdot 2 (3 - y^{3})}{y^{2}}$

Step 1.2.9

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

$\frac{d y}{d x} = \frac{2 \cdot x (3 - y^{3})}{y^{2}}$

Step 1.2.10

Multiply $2$ by $x$ .

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

Step 1.3

Regroup factors.

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

Step 1.4

Multiply both sides by $\frac{y^{2}}{3 - y^{3}}$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.2

Combine $2$ and $\frac{y^{2}}{3 - y^{3}}$ .

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

Step 1.5.3

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

$\frac{y^{2}}{3 - y^{3}} \frac{d y}{d x} = \frac{2 y^{2}}{3 - y^{3}} \cdot \frac{x (3 - y^{3})}{y^{2}}$

Step 1.5.4

Cancel the common factor of $y^{2}$ .

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Step 1.5.4.1

Factor $y^{2}$ out of $2 y^{2}$ .

$\frac{y^{2}}{3 - y^{3}} \frac{d y}{d x} = \frac{y^{2} \cdot 2}{3 - y^{3}} \cdot \frac{x (3 - y^{3})}{y^{2}}$

Step 1.5.4.2

Cancel the common factor.

$\frac{y^{2}}{3 - y^{3}} \frac{d y}{d x} = \frac{y^{2} \cdot 2}{3 - y^{3}} \cdot \frac{x (3 - y^{3})}{y^{2}}$

Step 1.5.4.3

Rewrite the expression.

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

Step 1.5.5

Cancel the common factor of $3 - y^{3}$ .

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Step 1.5.5.1

Factor $3 - y^{3}$ out of $x (3 - y^{3})$ .

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

Step 1.5.5.2

Cancel the common factor.

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

Step 1.5.5.3

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Rewrite $y^{3}$ as ${(y^{3})}^{1}$ .

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

Step 2.2.2

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

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Step 2.2.2.1

Let $u_{1} = y^{3}$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.2.1.1

Differentiate $y^{3}$ .

$\frac{d}{d y} [y^{3}]$

Step 2.2.2.1.2

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

$3 y^{2}$

Step 2.2.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{3 - {u_{1}}^{1}} \cdot \frac{1}{3} d u_{1} = \int 2 x d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify.

$\int \frac{1}{3 - u_{1}} \cdot \frac{1}{3} d u_{1} = \int 2 x d x$

Step 2.2.3.2

Multiply $\frac{1}{3 - u_{1}}$ by $\frac{1}{3}$ .

$\int \frac{1}{(3 - u_{1}) \cdot 3} d u_{1} = \int 2 x d x$

Step 2.2.3.3

Move $3$ to the left of $3 - u_{1}$ .

$\int \frac{1}{3 (3 - u_{1})} d u_{1} = \int 2 x d x$

Step 2.2.4

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

$\frac{1}{3} \int \frac{1}{3 - u_{1}} d u_{1} = \int 2 x d x$

Step 2.2.5

Let $u_{2} = 3 - u_{1}$ . Then $d u_{2} = - d u_{1}$ , so $- d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.2.5.1

Let $u_{2} = 3 - u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.2.5.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.5.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.5.2

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

$\frac{1}{3} \int \frac{- 1}{u_{2}} d u_{2} = \int 2 x d x$

Step 2.2.6

Move the negative in front of the fraction.

$\frac{1}{3} \int - \frac{1}{u_{2}} d u_{2} = \int 2 x d x$

Step 2.2.7

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\frac{1}{3} (- \int \frac{1}{u_{2}} d u_{2}) = \int 2 x d x$

Step 2.2.8

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{3} (- (\ln (| u_{2} |) + C_{1})) = \int 2 x d x$

Step 2.2.9

Simplify.

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Step 2.2.9.1

Simplify.

$\frac{1}{3} (- \ln (| u_{2} |)) + C_{1} = \int 2 x d x$

Step 2.2.9.2

Combine $\frac{1}{3}$ and $\ln (| u_{2} |)$ .

$- \frac{\ln (| u_{2} |)}{3} + C_{1} = \int 2 x d x$

Step 2.2.10

Substitute back in for each integration substitution variable.

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Step 2.2.10.1

Replace all occurrences of $u_{2}$ with $3 - u_{1}$ .

$- \frac{\ln (| 3 - u_{1} |)}{3} + C_{1} = \int 2 x d x$

Step 2.2.10.2

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

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

Step 2.2.11

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C_{2})$ as $2 (\frac{1}{2}) x^{2} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

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

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 $- 3 (- \frac{1}{3} \ln (| 3 - y^{3} |))$ .

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Step 3.2.1.1.1

Combine $\ln (| 3 - y^{3} |)$ and $\frac{1}{3}$ .

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Move the leading negative in $- \frac{\ln (| 3 - y^{3} |)}{3}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

$3 \cdot - 1 \frac{- \ln (| 3 - y^{3} |)}{3} = - 3 (x^{2} + C)$

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

Multiply $\ln (| 3 - y^{3} |)$ by $1$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Subtract $3$ from both sides of the equation.

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

Step 3.5.4

Divide each term in $- y^{3} = \pm e^{- 3 x^{2} - 3 C} - 3$ by $- 1$ and simplify.

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Step 3.5.4.1

Divide each term in $- y^{3} = \pm e^{- 3 x^{2} - 3 C} - 3$ by $- 1$ .

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.4.2.2

Divide $y^{3}$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Simplify each term.

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Step 3.5.4.3.1.1

Move the negative one from the denominator of $\frac{\pm e^{- 3 x^{2} - 3 C}}{- 1}$ .

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

Step 3.5.4.3.1.2

Rewrite $- 1 \cdot (\pm e^{- 3 x^{2} - 3 C})$ as $- (\pm e^{- 3 x^{2} - 3 C})$ .

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

Step 3.5.4.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 3.5.5

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

$y = \sqrt[3]{- (\pm e^{- 3 x^{2} - 3 C}) + 3}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \sqrt[3]{- (\pm e^{- 3 x^{2} + C}) + 3}$

Step 4.2

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

$y = \sqrt[3]{- (\pm e^{- 3 x^{2}} e^{C}) + 3}$

Step 4.3

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

$y = \sqrt[3]{- (\pm e^{C} e^{- 3 x^{2}}) + 3}$

Step 4.4

Combine constants with the plus or minus.

$y = \sqrt[3]{- (C e^{- 3 x^{2}}) + 3}$