Calculus Examples

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

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

Add $y$ to both sides of the equation.

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

Step 1.2

Reorder terms.

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

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.

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

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

Step 6.2

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

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

Step 6.3

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

$x y = 6 (\frac{1}{3} x^{3} + C) + \int 2 e^{x} x d x$

Step 6.4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$x y = 6 (\frac{1}{3} x^{3} + C) + 2 \int e^{x} x d x$

Step 6.5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$x y = 6 (\frac{1}{3} x^{3} + C) + 2 (x e^{x} - \int e^{x} d x)$

Step 6.6

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$x y = 6 (\frac{1}{3} x^{3} + C) + 2 (x e^{x} - (e^{x} + C))$

Step 6.7

Simplify.

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Step 6.7.1

Simplify.

$x y = 6 (\frac{1}{3}) x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2

Simplify.

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Step 6.7.2.1

Combine $6$ and $\frac{1}{3}$ .

$x y = \frac{6}{3} x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2.2

Cancel the common factor of $6$ and $3$ .

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Step 6.7.2.2.1

Factor $3$ out of $6$ .

$x y = \frac{3 \cdot 2}{3} x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2.2.2

Cancel the common factors.

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Step 6.7.2.2.2.1

Factor $3$ out of $3$ .

$x y = \frac{3 \cdot 2}{3 (1)} x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2.2.2.2

Cancel the common factor.

$x y = \frac{3 \cdot 2}{3 \cdot 1} x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2.2.2.3

Rewrite the expression.

$x y = \frac{2}{1} x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 6.7.2.2.2.4

Divide $2$ by $1$ .

$x y = 2 x^{3} + 2 (x e^{x} - e^{x}) + C$

Step 7

Divide each term in $x y = 2 x^{3} + 2 (x e^{x} - e^{x}) + C$ by $x$ and simplify.

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Step 7.1

Divide each term in $x y = 2 x^{3} + 2 (x e^{x} - e^{x}) + C$ by $x$ .

$\frac{x y}{x} = \frac{2 x^{3}}{x} + \frac{2 (x e^{x} - e^{x})}{x} + \frac{C}{x}$

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.

$\frac{x y}{x} = \frac{2 x^{3}}{x} + \frac{2 (x e^{x} - e^{x})}{x} + \frac{C}{x}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x^{3}}{x} + \frac{2 (x e^{x} - e^{x})}{x} + \frac{C}{x}$

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 $x^{3}$ and $x$ .

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Step 7.3.1.1.1

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

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

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

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

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

Step 7.3.1.1.2.2

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

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

Step 7.3.1.1.2.3

Cancel the common factor.

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

Step 7.3.1.1.2.4

Rewrite the expression.

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

Step 7.3.1.1.2.5

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

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

Step 7.3.1.2

Factor $e^{x}$ out of $x e^{x} - e^{x}$ .

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Step 7.3.1.2.1

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

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

Step 7.3.1.2.2

Factor $e^{x}$ out of $- e^{x}$ .

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

Step 7.3.1.2.3

Factor $e^{x}$ out of $e^{x} x + e^{x} \cdot - 1$ .

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

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

Step 7.3.2

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

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

Step 7.3.3

Combine the numerators over the common denominator.

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.3.5.1

Move $x$ .

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

Step 7.3.5.2

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

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Step 7.3.5.2.1

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

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

Step 7.3.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.3.5.3

Add $1$ and $2$ .

$y = \frac{2 x^{3} + 2 e^{x} (x - 1) + C}{x}$

Step 7.3.6

Simplify the numerator.

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Step 7.3.6.1

Apply the distributive property.

$y = \frac{2 x^{3} + 2 e^{x} x + 2 e^{x} \cdot - 1 + C}{x}$

Step 7.3.6.2

Multiply $- 1$ by $2$ .

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

Step 7.3.7

Reorder factors in $\frac{2 x^{3} + 2 e^{x} x - 2 e^{x} + C}{x}$ .

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