Calculus Examples

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

Step 1

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

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Remove parentheses.

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

Step 1.6

Move $y$ .

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

Step 2

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

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

Step 3

Set up an integral on each side.

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

Step 4

Integrate the left side.

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

Step 5

Integrate the right side.

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Step 5.1

Split the single integral into multiple integrals.

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

Step 5.2

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

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

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $y$ by $1$ .

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

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

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

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Step 6.3.1.1.1

Cancel the common factor.

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

Step 6.3.1.1.2

Divide $\frac{1}{2}$ by $1$ .

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

Step 6.3.1.2

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

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Step 6.3.1.2.1

Factor $x$ out of $- x$ .

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

Step 6.3.1.2.2

Cancel the common factors.

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Step 6.3.1.2.2.1

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

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

Step 6.3.1.2.2.2

Cancel the common factor.

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

Step 6.3.1.2.2.3

Rewrite the expression.

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

Step 6.3.1.3

Move the negative in front of the fraction.

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