Calculus Examples

$x \frac{d y}{d x} + 9 y + 6 = 6 x^{2} + \frac{d y}{d x} + 8 y$

Step 1

Rewrite the equation as $M (x) \frac{d y}{d x} + P (x) y = Q (x)$ .

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Step 1.1

Subtract $8 y$ from both sides of the equation.

$x \frac{d y}{d x} + 9 y + 6 - 8 y = 6 x^{2} + \frac{d y}{d x}$

Step 1.2

Subtract $6$ from both sides of the equation.

$x \frac{d y}{d x} + 9 y - 8 y = 6 x^{2} + \frac{d y}{d x} - 6$

Step 1.3

Subtract $8 y$ from $9 y$ .

$x \frac{d y}{d x} + y = 6 x^{2} + \frac{d y}{d x} - 6$

Step 1.4

Reorder terms.

$x \frac{d y}{d x} + y = \frac{d y}{d x} + 6 x^{2} - 6$

Step 2

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

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Step 2.1

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 2.2

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

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

Step 2.3

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 2.4

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

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

Step 2.5

Reorder $y$ and $1$ .

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

Step 2.6

Multiply $y$ by $1$ .

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

Step 3

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

$\frac{d}{d x} [x y] = \frac{d y}{d x} + 6 x^{2} - 6$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int \frac{d y}{d x} + 6 x^{2} - 6 d x$

Step 5

Integrate the left side.

$x y = \int \frac{d y}{d x} + 6 x^{2} - 6 d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$x y = \int \frac{d y}{d x} d x + \int 6 x^{2} d x + \int - 6 d x$

Step 6.2

Apply the constant rule.

$x y = \frac{d y}{d x} x + C + \int 6 x^{2} d x + \int - 6 d x$

Step 6.3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$x y = \frac{d y}{d x} x + C + 6 \int x^{2} d x + \int - 6 d x$

Step 6.4

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

$x y = \frac{d y}{d x} x + C + 6 (\frac{1}{3} x^{3} + C) + \int - 6 d x$

Step 6.5

Apply the constant rule.

$x y = \frac{d y}{d x} x + C + 6 (\frac{1}{3} x^{3} + C) - 6 x + C$

Step 6.6

Simplify.

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Step 6.6.1

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

$x y = \frac{d y}{d x} x + C + 6 (\frac{x^{3}}{3} + C) - 6 x + C$

Step 6.6.2

Simplify.

$x y = \frac{d y}{d x} x + 2 x^{3} - 6 x + C$

Step 7

Divide each term in $x y = \frac{d y}{d x} x + 2 x^{3} - 6 x + C$ by $x$ and simplify.

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Step 7.1

Divide each term in $x y = \frac{d y}{d x} x + 2 x^{3} - 6 x + C$ by $x$ .

$\frac{x y}{x} = \frac{\frac{d y}{d x} x}{x} + \frac{2 x^{3}}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $x$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{\frac{d y}{d x} x}{x} + \frac{2 x^{3}}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{d y}{d x} x}{x} + \frac{2 x^{3}}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $x$ .

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Step 7.3.1.1.1

Cancel the common factor.

$y = \frac{\frac{d y}{d x} x}{x} + \frac{2 x^{3}}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.1.2

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

$y = \frac{d y}{d x} + \frac{2 x^{3}}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2

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

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Step 7.3.1.2.1

Factor $x$ out of $2 x^{3}$ .

$y = \frac{d y}{d x} + \frac{x (2 x^{2})}{x} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2.2

Cancel the common factors.

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Step 7.3.1.2.2.1

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

$y = \frac{d y}{d x} + \frac{x (2 x^{2})}{x^{1}} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2.2.2

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

$y = \frac{d y}{d x} + \frac{x (2 x^{2})}{x \cdot 1} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2.2.3

Cancel the common factor.

$y = \frac{d y}{d x} + \frac{x (2 x^{2})}{x \cdot 1} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2.2.4

Rewrite the expression.

$y = \frac{d y}{d x} + \frac{2 x^{2}}{1} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.2.2.5

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

$y = \frac{d y}{d x} + 2 x^{2} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.3

Cancel the common factor of $x$ .

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Step 7.3.1.3.1

Cancel the common factor.

$y = \frac{d y}{d x} + 2 x^{2} + \frac{- 6 x}{x} + \frac{C}{x}$

Step 7.3.1.3.2

Divide $- 6$ by $1$ .

$y = \frac{d y}{d x} + 2 x^{2} - 6 + \frac{C}{x}$