Calculus Examples

$x^{2} \frac{d y}{d x} + 2 x^{3} y = 4 x^{3}$

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

$x^{2} \frac{d y}{d x} = 4 x^{3} - 2 x^{3} y$

Step 1.1.2

Divide each term in $x^{2} \frac{d y}{d x} = 4 x^{3} - 2 x^{3} y$ by $x^{2}$ and simplify.

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Step 1.1.2.1

Divide each term in $x^{2} \frac{d y}{d x} = 4 x^{3} - 2 x^{3} y$ by $x^{2}$ .

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{4 x^{3}}{x^{2}} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

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

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{4 x^{3}}{x^{2}} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.2.1.2

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

$\frac{d y}{d x} = \frac{4 x^{3}}{x^{2}} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

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Step 1.1.2.3.1.1.1

Factor $x^{2}$ out of $4 x^{3}$ .

$\frac{d y}{d x} = \frac{x^{2} (4 x)}{x^{2}} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.2.3.1.1.2.1

Multiply by $1$ .

$\frac{d y}{d x} = \frac{x^{2} (4 x)}{x^{2} \cdot 1} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x^{2} (4 x)}{x^{2} \cdot 1} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{4 x}{1} + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3.1.1.2.4

Divide $4 x$ by $1$ .

$\frac{d y}{d x} = 4 x + \frac{- 2 x^{3} y}{x^{2}}$

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

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

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

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Multiply by $1$ .

$\frac{d y}{d x} = 4 x + \frac{x^{2} (- 2 x y)}{x^{2} \cdot 1}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = 4 x + \frac{x^{2} (- 2 x y)}{x^{2} \cdot 1}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = 4 x + \frac{- 2 x y}{1}$

Step 1.1.2.3.1.2.2.4

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

$\frac{d y}{d x} = 4 x - 2 x y$

Step 1.2

Factor.

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Step 1.2.1

Factor $2 x$ out of $4 x - 2 x y$ .

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Step 1.2.1.1

Factor $2 x$ out of $4 x$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Factor $2 x$ out of $2 x (2) + 2 x (- 1 y)$ .

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2

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

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

Step 1.4.3

Cancel the common factor of $2 - y$ .

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Step 1.4.3.1

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

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

Step 1.4.3.2

Cancel the common factor.

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

Step 1.4.3.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 2 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.1.2

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

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

Step 2.2.2

Split the fraction into multiple fractions.

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

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

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

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.

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

Step 2.3.2

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

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

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

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.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- \ln (| 2 - y |) = x^{2} + C$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \ln (| 2 - y |) = x^{2} + C$ by $- 1$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.2.2

Divide $\ln (| 2 - y |)$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

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

Step 3.1.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

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

Step 3.1.3.1.3

Move the negative one from the denominator of $\frac{C}{- 1}$ .

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

Step 3.1.3.1.4

Rewrite $- 1 \cdot C$ as $- C$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Subtract $2$ from both sides of the equation.

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

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Dividing two negative values results in a positive value.

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

Step 3.4.4.2.2

Divide $y$ by $1$ .

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

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Simplify each term.

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Step 3.4.4.3.1.1

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

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

Step 3.4.4.3.1.2

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

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

Step 3.4.4.3.1.3

Divide $- 2$ by $- 1$ .

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

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine constants with the plus or minus.

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