Calculus Examples

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

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

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.1.3

Move $- 2$ to the left of $x^{2}$ .

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

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Add $2 x^{2}$ to both sides of the equation.

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Multiply both sides by $\frac{1}{- 3 y + 2}$ .

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

Step 1.4

Cancel the common factor of $- 3 y + 2$ .

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Differentiate $- 3 y + 2$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [- 3 y]$ .

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Step 2.2.1.1.3.1

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

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

Step 2.2.1.1.3.2

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

$- 3 \cdot 1 + \frac{d}{d y} [2]$

Step 2.2.1.1.3.3

Multiply $- 3$ by $1$ .

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

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

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

$- 3 + 0$

Step 2.2.1.1.4.2

Add $- 3$ and $0$ .

$- 3$

Step 2.2.1.2

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2.2

Multiply $\frac{1}{u}$ by $\frac{1}{3}$ .

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

Step 2.2.2.3

Move $3$ to the left of $u$ .

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

Step 2.2.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Replace all occurrences of $u$ with $- 3 y + 2$ .

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

Step 2.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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 $- 3 (- \frac{1}{3} \ln (| - 3 y + 2 |))$ .

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Step 3.2.1.1.1

Combine $\ln (| - 3 y + 2 |)$ and $\frac{1}{3}$ .

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

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 $- \frac{\ln (| - 3 y + 2 |)}{3}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

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

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

Multiply $\ln (| - 3 y + 2 |)$ by $1$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 3 (\frac{1}{3} x^{3} + C)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1

Factor $3$ out of $- 3$ .

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

Step 3.2.2.1.3.2

Cancel the common factor.

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

Step 3.2.2.1.3.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Subtract $2$ from both sides of the equation.

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

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $- 3$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

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

Simplify $\frac{\pm e^{- x^{3} - 3 C}}{- 3}$ .

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

Step 3.5.4.3.1.2

Dividing two negative values results in a positive value.

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

Step 3.5.4.3.2

Combine the numerators over the common denominator.

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

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine constants with the plus or minus.

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