Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $x y$ .

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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 $f (x) = x$ and $g (x) = y$ .

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

Step 1.2

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

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

Step 1.3

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

$x y' + y \cdot 1$

Step 1.4

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

$x \frac{d y}{d x} + y \cdot 1$

Step 1.5

Reorder $y$ and $1$ .

$x \frac{d y}{d x} + 1 \cdot y$

Step 1.6

Multiply $y$ by $1$ .

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

Step 2

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

$\frac{d}{d x} [x y] = 3 x^{2}$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int 3 x^{2} d x$

Step 4

Integrate the left side.

$x y = \int 3 x^{2} d x$

Step 5

Integrate the right side.

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Step 5.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$x y = 3 \int x^{2} d x$

Step 5.2

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

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

Step 5.3

Simplify the answer.

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Step 5.3.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

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

Step 5.3.2

Simplify.

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Step 5.3.2.1

Combine $3$ and $\frac{1}{3}$ .

$x y = \frac{3}{3} x^{3} + C$

Step 5.3.2.2

Cancel the common factor of $3$ .

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Step 5.3.2.2.1

Cancel the common factor.

$x y = \frac{3}{3} x^{3} + C$

Step 5.3.2.2.2

Rewrite the expression.

$x y = 1 x^{3} + C$

Step 5.3.2.3

Multiply $x^{3}$ by $1$ .

$x y = x^{3} + C$

Step 6

Divide each term in $x y = x^{3} + C$ by $x$ and simplify.

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Step 6.1

Divide each term in $x y = x^{3} + C$ by $x$ .

$\frac{x y}{x} = \frac{x^{3}}{x} + \frac{C}{x}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $x$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{x^{3}}{x} + \frac{C}{x}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{x} + \frac{C}{x}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

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

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

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Raise $x$ to the power of $1$ .

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

Step 6.3.1.2.2

Factor $x$ out of $x^{1}$ .

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

Step 6.3.1.2.3

Cancel the common factor.

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

Step 6.3.1.2.4

Rewrite the expression.

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

Step 6.3.1.2.5

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

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