Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $(x^{2} - 1) y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} - 1$ and $g (x) = y$ .

$(x^{2} - 1) \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2} - 1]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$(x^{2} - 1) y' + y \frac{d}{d x} [x^{2} - 1]$

Step 1.3

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$(x^{2} - 1) y' + y (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{2} - 1) y' + y (2 x + \frac{d}{d x} [- 1])$

Step 1.5

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

$(x^{2} - 1) y' + y (2 x + 0)$

Step 1.6

Add $2 x$ and $0$ .

$(x^{2} - 1) y' + y (2 x)$

Step 1.7

Substitute $\frac{d y}{d x}$ for $y'$ .

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

Step 1.8

Remove parentheses.

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

Step 1.9

Move $y$ .

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

Step 2

Rewrite the left side as a result of differentiating a product.

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

Step 3

Set up an integral on each side.

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

Step 4

Integrate the left side.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $x^{2} - 1$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $y$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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

Step 6.3.1.2

Simplify the denominator.

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Step 6.3.1.2.1

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

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

Step 6.3.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 6.3.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3.1.4

Combine.

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

Step 6.3.1.5

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

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

Step 6.3.1.6

Simplify the denominator.

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Step 6.3.1.6.1

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

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

Step 6.3.1.6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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