Calculus Examples

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

Step 1

Let $v = \frac{d y}{d x}$ . Then $\frac{d v}{d x} = \frac{d^{2} y}{d x^{2}}$ . Substitute $v$ for $\frac{d y}{d x}$ and $\frac{d v}{d x}$ for $\frac{d^{2} y}{d x^{2}}$ to get a differential equation with dependent variable $v$ and independent variable $x$ .

$x \frac{d v}{d x} + 2 v = 6 x$

Step 2

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

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Step 2.1

Divide each term in $x \frac{d v}{d x} + 2 v = 6 x$ by $x$ .

$\frac{x \frac{d v}{d x}}{x} + \frac{2 v}{x} = \frac{6 x}{x}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\frac{x \frac{d v}{d x}}{x} + \frac{2 v}{x} = \frac{6 x}{x}$

Step 2.2.2

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

$\frac{d v}{d x} + \frac{2 v}{x} = \frac{6 x}{x}$

Step 2.3

Cancel the common factor of $x$ .

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Step 2.3.1

Cancel the common factor.

$\frac{d v}{d x} + \frac{2 v}{x} = \frac{6 x}{x}$

Step 2.3.2

Divide $6$ by $1$ .

$\frac{d v}{d x} + \frac{2 v}{x} = 6$

Step 2.4

Factor $v$ out of $\frac{2 v}{x}$ .

$\frac{d v}{d x} + v \frac{2}{x} = 6$

Step 2.5

Reorder $v$ and $\frac{2}{x}$ .

$\frac{d v}{d x} + \frac{2}{x} v = 6$

Step 3

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{x}$ .

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Step 3.1

Set up the integration.

$e^{\int \frac{2}{x} d x}$

Step 3.2

Integrate $\frac{2}{x}$ .

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Step 3.2.1

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

$e^{2 \int \frac{1}{x} d x}$

Step 3.2.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$e^{2 (\ln (| x |) + C_{1})}$

Step 3.2.3

Simplify.

$e^{2 \ln (| x |) + C_{1}}$

Step 3.3

Remove the constant of integration.

$e^{2 \ln (x)}$

Step 3.4

Use the logarithmic power rule.

$e^{\ln (x^{2})}$

Step 3.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 4

Multiply each term by the integrating factor $x^{2}$ .

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Step 4.1

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

$x^{2} \frac{d v}{d x} + x^{2} (\frac{2}{x} v) = x^{2} \cdot 6$

Step 4.2

Simplify each term.

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Step 4.2.1

Combine $\frac{2}{x}$ and $v$ .

$x^{2} \frac{d v}{d x} + x^{2} \frac{2 v}{x} = x^{2} \cdot 6$

Step 4.2.2

Cancel the common factor of $x$ .

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Step 4.2.2.1

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

$x^{2} \frac{d v}{d x} + x \cdot x \frac{2 v}{x} = x^{2} \cdot 6$

Step 4.2.2.2

Cancel the common factor.

$x^{2} \frac{d v}{d x} + x \cdot x \frac{2 v}{x} = x^{2} \cdot 6$

Step 4.2.2.3

Rewrite the expression.

$x^{2} \frac{d v}{d x} + x (2 v) = x^{2} \cdot 6$

Step 4.2.3

Rewrite using the commutative property of multiplication.

$x^{2} \frac{d v}{d x} + 2 x v = x^{2} \cdot 6$

Step 4.3

Move $6$ to the left of $x^{2}$ .

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

Step 5

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

$\frac{d}{d x} [x^{2} v] = 6 x^{2}$

Step 6

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} v] d x = \int 6 x^{2} d x$

Step 7

Integrate the left side.

$x^{2} v = \int 6 x^{2} d x$

Step 8

Integrate the right side.

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Step 8.1

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

$x^{2} v = 6 \int x^{2} d x$

Step 8.2

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

$x^{2} v = 6 (\frac{1}{3} x^{3} + C_{2})$

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $6 (\frac{1}{3} x^{3} + C_{2})$ as $6 (\frac{1}{3}) x^{3} + C_{2}$ .

$x^{2} v = 6 (\frac{1}{3}) x^{3} + C_{2}$

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

$x^{2} v = \frac{6}{3} x^{3} + C_{2}$

Step 8.3.2.2

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

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Step 8.3.2.2.1

Factor $3$ out of $6$ .

$x^{2} v = \frac{3 \cdot 2}{3} x^{3} + C_{2}$

Step 8.3.2.2.2

Cancel the common factors.

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Step 8.3.2.2.2.1

Factor $3$ out of $3$ .

$x^{2} v = \frac{3 \cdot 2}{3 (1)} x^{3} + C_{2}$

Step 8.3.2.2.2.2

Cancel the common factor.

$x^{2} v = \frac{3 \cdot 2}{3 \cdot 1} x^{3} + C_{2}$

Step 8.3.2.2.2.3

Rewrite the expression.

$x^{2} v = \frac{2}{1} x^{3} + C_{2}$

Step 8.3.2.2.2.4

Divide $2$ by $1$ .

$x^{2} v = 2 x^{3} + C_{2}$

Step 9

Divide each term in $x^{2} v = 2 x^{3} + C_{2}$ by $x^{2}$ and simplify.

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Step 9.1

Divide each term in $x^{2} v = 2 x^{3} + C_{2}$ by $x^{2}$ .

$\frac{x^{2} v}{x^{2}} = \frac{2 x^{3}}{x^{2}} + \frac{C_{2}}{x^{2}}$

Step 9.2

Simplify the left side.

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Step 9.2.1

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

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Step 9.2.1.1

Cancel the common factor.

$\frac{x^{2} v}{x^{2}} = \frac{2 x^{3}}{x^{2}} + \frac{C_{2}}{x^{2}}$

Step 9.2.1.2

Divide $v$ by $1$ .

$v = \frac{2 x^{3}}{x^{2}} + \frac{C_{2}}{x^{2}}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 9.3.1.1

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

$v = \frac{x^{2} (2 x)}{x^{2}} + \frac{C_{2}}{x^{2}}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Multiply by $1$ .

$v = \frac{x^{2} (2 x)}{x^{2} \cdot 1} + \frac{C_{2}}{x^{2}}$

Step 9.3.1.2.2

Cancel the common factor.

$v = \frac{x^{2} (2 x)}{x^{2} \cdot 1} + \frac{C_{2}}{x^{2}}$

Step 9.3.1.2.3

Rewrite the expression.

$v = \frac{2 x}{1} + \frac{C_{2}}{x^{2}}$

Step 9.3.1.2.4

Divide $2 x$ by $1$ .

$v = 2 x + \frac{C_{2}}{x^{2}}$

Step 10

Replace all occurrences of $v$ with $\frac{d y}{d x}$ .

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

Step 11

Rewrite the equation.

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

Step 12

Integrate both sides.

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Step 12.1

Set up an integral on each side.

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

Step 12.2

Apply the constant rule.

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

Step 12.3

Integrate the right side.

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Step 12.3.1

Split the single integral into multiple integrals.

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

Step 12.3.2

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

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

Step 12.3.3

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

$y + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4}) + \int \frac{C_{2}}{x^{2}} d x$

Step 12.3.4

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

$y + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4}) + C_{2} \int \frac{1}{x^{2}} d x$

Step 12.3.5

Simplify the expression.

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Step 12.3.5.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4}) + C_{2} \int {(x^{2})}^{- 1} d x$

Step 12.3.5.2

Simplify.

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Step 12.3.5.2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) + C_{2} \int {(x^{2})}^{- 1} d x$

Step 12.3.5.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 12.3.5.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) + C_{2} \int x^{2 \cdot - 1} d x$

Step 12.3.5.2.2.2

Multiply $2$ by $- 1$ .

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) + C_{2} \int x^{- 2} d x$

Step 12.3.6

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

$y + C_{3} = 2 (\frac{x^{2}}{2} + C_{4}) + C_{2} (- x^{- 1} + C_{5})$

Step 12.3.7

Simplify.

$y + C_{3} = x^{2} + C_{2} (- x^{- 1}) + C_{6}$

Step 12.3.8

Reorder terms.

$y + C_{3} = x^{2} - C_{2} x^{- 1} + C_{6}$

Step 12.4

Group the constant of integration on the right side as $D$ .

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