Calculus Examples

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

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.

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

Step 1.1.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.1.3

Apply the distributive property.

$y \frac{d y}{d x} - 1 x^{2} - y x^{2} = 0$

Step 1.1.1.4

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

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

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

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $y$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

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

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Cancel the common factor of $y$ .

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Step 1.1.3.3.1.1

Cancel the common factor.

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

Step 1.1.3.3.1.2

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

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

Step 1.2

Factor.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Multiply by $1$ .

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

Step 1.2.1.3

Factor $x^{2}$ out of $x^{2} \frac{1}{y} + x^{2} \cdot 1$ .

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

Step 1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Combine $x^{2}$ and $\frac{1 + y}{y}$ .

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

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.4.3

Cancel the common factor of $1 + y$ .

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Step 1.4.3.1

Factor $1 + y$ out of $x^{2} (1 + y)$ .

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

Step 1.4.3.2

Cancel the common factor.

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

Step 1.4.3.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Reorder $1$ and $y$ .

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

Step 2.2.2

Divide $y$ by $y + 1$ .

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Step 2.2.2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y$

$1$

$y$

$0$

Step 2.2.2.2

Divide the highest order term in the dividend $y$ by the highest order term in divisor $y$ .

					$1$
	$y$	+	$1$		$y$	+	$0$

Step 2.2.2.3

Multiply the new quotient term by the divisor.

				$1$
$y$	+	$1$		$y$	+	$0$
			+	$y$	+	$1$

Step 2.2.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $y + 1$

				$1$
$y$	+	$1$		$y$	+	$0$
			-	$y$	-	$1$

Step 2.2.2.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$1$
$y$	+	$1$		$y$	+	$0$
			-	$y$	-	$1$
					-	$1$

Step 2.2.2.6

The final answer is the quotient plus the remainder over the divisor.

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

Step 2.2.3

Split the single integral into multiple integrals.

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

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

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

Step 2.2.6

Let $u = y + 1$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.6.1

Let $u = y + 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.6.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.6.1.2

By the Sum Rule, the derivative of $y + 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.6.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.6.1.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.6.1.5

Add $1$ and $0$ .

$1$

Step 2.2.6.2

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify.

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

Step 2.2.9

Replace all occurrences of $u$ with $y + 1$ .

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

Step 2.3

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

$y - \ln (| y + 1 |) + C_{3} = \frac{1}{3} x^{3} + C_{4}$

Step 2.4

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

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