Calculus Examples

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

Step 1

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

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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 $f (x) = 1 + x^{2}$ and $g (x) = y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$(1 + x^{2}) y' + y (0 + \frac{d}{d x} [x^{2}])$

Step 1.5

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

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

Step 1.6

Add $0$ and $2 x$ .

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

Step 1.7

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

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

Step 1.8

Reorder $1$ and $x^{2}$ .

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

Step 1.9

Remove parentheses.

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

Step 1.10

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.

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

Step 3

Set up an integral on each side.

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

Step 4

Integrate the left side.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $y$ by $1$ .

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

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 $x^{2}$ .

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

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3.1.3

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{1 + x^{2}}$ .

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

Step 6.3.2

To write $\frac{C}{1 + x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.3.3

Write each expression with a common denominator of $2 (1 + x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.3.1

Multiply $\frac{C}{1 + x^{2}}$ by $\frac{2}{2}$ .

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

Step 6.3.3.2

Reorder the factors of $(1 + x^{2}) \cdot 2$ .

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

Step 6.3.4

Combine the numerators over the common denominator.

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

Step 6.3.5

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

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