Calculus Examples

$(x - 34) \frac{d y}{d x} - y = {(x - 34)}^{3}$

Step 1

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

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Step 1.1

Divide each term in $(x - 34) \frac{d y}{d x} - y = {(x - 34)}^{3}$ by $x - 34$ .

$\frac{(x - 34) \frac{d y}{d x}}{x - 34} + \frac{- y}{x - 34} = \frac{{(x - 34)}^{3}}{x - 34}$

Step 1.2

Cancel the common factor of $x - 34$ .

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Step 1.2.1

Cancel the common factor.

$\frac{(x - 34) \frac{d y}{d x}}{x - 34} + \frac{- y}{x - 34} = \frac{{(x - 34)}^{3}}{x - 34}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{- y}{x - 34} = \frac{{(x - 34)}^{3}}{x - 34}$

Step 1.3

Cancel the common factor of ${(x - 34)}^{3}$ and $x - 34$ .

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Step 1.3.1

Factor $x - 34$ out of ${(x - 34)}^{3}$ .

$\frac{d y}{d x} + \frac{- y}{x - 34} = \frac{(x - 34) {(x - 34)}^{2}}{x - 34}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Multiply by $1$ .

$\frac{d y}{d x} + \frac{- y}{x - 34} = \frac{(x - 34) {(x - 34)}^{2}}{(x - 34) \cdot 1}$

Step 1.3.2.2

Cancel the common factor.

$\frac{d y}{d x} + \frac{- y}{x - 34} = \frac{(x - 34) {(x - 34)}^{2}}{(x - 34) \cdot 1}$

Step 1.3.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- y}{x - 34} = \frac{{(x - 34)}^{2}}{1}$

Step 1.3.2.4

Divide ${(x - 34)}^{2}$ by $1$ .

$\frac{d y}{d x} + \frac{- y}{x - 34} = {(x - 34)}^{2}$

Step 1.4

Factor $y$ out of $\frac{- y}{x - 34}$ .

$\frac{d y}{d x} + y \frac{- 1}{x - 34} = {(x - 34)}^{2}$

Step 1.5

Reorder $y$ and $\frac{- 1}{x - 34}$ .

$\frac{d y}{d x} + \frac{- 1}{x - 34} y = {(x - 34)}^{2}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{x - 34}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 1}{x - 34} d x}$

Step 2.2

Integrate $\frac{- 1}{x - 34}$ .

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Step 2.2.1

Move the negative in front of the fraction.

$e^{\int - \frac{1}{x - 34} d x}$

Step 2.2.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{- \int \frac{1}{x - 34} d x}$

Step 2.2.3

Let $u = x - 34$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.3.1

Let $u = x - 34$ . Find $\frac{d u}{d x}$ .

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Step 2.2.3.1.1

Differentiate $x - 34$ .

$\frac{d}{d x} [x - 34]$

Step 2.2.3.1.2

By the Sum Rule, the derivative of $x - 34$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 34]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 34]$

Step 2.2.3.1.3

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

$1 + \frac{d}{d x} [- 34]$

Step 2.2.3.1.4

Since $- 34$ is constant with respect to $x$ , the derivative of $- 34$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.2.3.1.5

Add $1$ and $0$ .

$1$

Step 2.2.3.2

Rewrite the problem using $u$ and $d u$ .

$e^{- \int \frac{1}{u} d u}$

Step 2.2.4

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

$e^{- (\ln (| u |) + C)}$

Step 2.2.5

Simplify.

$e^{- \ln (| u |) + C}$

Step 2.2.6

Replace all occurrences of $u$ with $x - 34$ .

$e^{- \ln (| x - 34 |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- \ln (x - 34)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln ({(x - 34)}^{- 1})}$

Step 2.5

Exponentiation and log are inverse functions.

${(x - 34)}^{- 1}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x - 34}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x - 34}$ .

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Step 3.1

Multiply each term by $\frac{1}{x - 34}$ .

$\frac{1}{x - 34} \frac{d y}{d x} + \frac{1}{x - 34} (\frac{- 1}{x - 34} y) = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x - 34}$ and $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{x - 34} + \frac{1}{x - 34} (\frac{- 1}{x - 34} y) = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.2

Move the negative in front of the fraction.

$\frac{\frac{d y}{d x}}{x - 34} + \frac{1}{x - 34} (- \frac{1}{x - 34} y) = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x - 34} - \frac{1}{x - 34} (\frac{1}{x - 34} y) = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.4

Combine $\frac{1}{x - 34}$ and $y$ .

$\frac{\frac{d y}{d x}}{x - 34} - \frac{1}{x - 34} \cdot \frac{y}{x - 34} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.5

Multiply $- \frac{1}{x - 34} \cdot \frac{y}{x - 34}$ .

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Step 3.2.5.1

Multiply $\frac{y}{x - 34}$ by $\frac{1}{x - 34}$ .

$\frac{\frac{d y}{d x}}{x - 34} - \frac{y}{(x - 34) (x - 34)} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.5.2

Raise $x - 34$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{x - 34} - \frac{y}{{(x - 34)}^{1} (x - 34)} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.5.3

Raise $x - 34$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{x - 34} - \frac{y}{{(x - 34)}^{1} {(x - 34)}^{1}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.5.4

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

$\frac{\frac{d y}{d x}}{x - 34} - \frac{y}{{(x - 34)}^{1 + 1}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.2.5.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d x}}{x - 34} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.3

To write $\frac{\frac{d y}{d x}}{x - 34}$ as a fraction with a common denominator, multiply by $\frac{x - 34}{x - 34}$ .

$\frac{\frac{d y}{d x}}{x - 34} \cdot \frac{x - 34}{x - 34} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.4

Write each expression with a common denominator of ${(x - 34)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{\frac{d y}{d x}}{x - 34}$ by $\frac{x - 34}{x - 34}$ .

$\frac{\frac{d y}{d x} (x - 34)}{(x - 34) (x - 34)} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.4.2

Raise $x - 34$ to the power of $1$ .

$\frac{\frac{d y}{d x} (x - 34)}{{(x - 34)}^{1} (x - 34)} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.4.3

Raise $x - 34$ to the power of $1$ .

$\frac{\frac{d y}{d x} (x - 34)}{{(x - 34)}^{1} {(x - 34)}^{1}} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.4.4

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

$\frac{\frac{d y}{d x} (x - 34)}{{(x - 34)}^{1 + 1}} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.4.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d x} (x - 34)}{{(x - 34)}^{2}} - \frac{y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{\frac{d y}{d x} (x - 34) - y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{\frac{d y}{d x} x + \frac{d y}{d x} \cdot - 34 - y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.6.2

Move $- 34$ to the left of $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x} x - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = \frac{1}{x - 34} {(x - 34)}^{2}$

Step 3.7

Cancel the common factor of $x - 34$ .

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Step 3.7.1

Factor $x - 34$ out of ${(x - 34)}^{2}$ .

$\frac{\frac{d y}{d x} x - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = \frac{1}{x - 34} ((x - 34) (x - 34))$

Step 3.7.2

Cancel the common factor.

$\frac{\frac{d y}{d x} x - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = \frac{1}{x - 34} ((x - 34) (x - 34))$

Step 3.7.3

Rewrite the expression.

$\frac{\frac{d y}{d x} x - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = x - 34$

Step 3.8

Reorder factors in $\frac{\frac{d y}{d x} x - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = x - 34$ .

$\frac{x \frac{d y}{d x} - 34 \frac{d y}{d x} - y}{{(x - 34)}^{2}} = x - 34$

Step 4

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

$\frac{d}{d x} [\frac{1}{x - 34} y] = x - 34$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x - 34} y] d x = \int x - 34 d x$

Step 6

Integrate the left side.

$\frac{1}{x - 34} y = \int x - 34 d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$\frac{1}{x - 34} y = \int x d x + \int - 34 d x$

Step 7.2

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

$\frac{1}{x - 34} y = \frac{1}{2} x^{2} + C + \int - 34 d x$

Step 7.3

Apply the constant rule.

$\frac{1}{x - 34} y = \frac{1}{2} x^{2} + C - 34 x + C$

Step 7.4

Simplify.

$\frac{1}{x - 34} y = \frac{1}{2} x^{2} - 34 x + C$

Step 8

Solve for $y$ .

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Step 8.1

Combine $\frac{1}{x - 34}$ and $y$ .

$\frac{y}{x - 34} = \frac{1}{2} x^{2} - 34 x + C$

Step 8.2

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

$\frac{y}{x - 34} = \frac{x^{2}}{2} - 34 x + C$

Step 8.3

Multiply both sides by $x - 34$ .

$\frac{y}{x - 34} (x - 34) = (\frac{x^{2}}{2} - 34 x + C) (x - 34)$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

Cancel the common factor of $x - 34$ .

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x - 34} (x - 34) = (\frac{x^{2}}{2} - 34 x + C) (x - 34)$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\frac{x^{2}}{2} - 34 x + C) (x - 34)$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{x^{2}}{2} - 34 x + C) (x - 34)$ .

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Step 8.4.2.1.1

Expand $(\frac{x^{2}}{2} - 34 x + C) (x - 34)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{x^{2}}{2} x + \frac{x^{2}}{2} \cdot - 34 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2

Simplify terms.

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Step 8.4.2.1.2.1

Simplify each term.

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Step 8.4.2.1.2.1.1

Multiply $\frac{x^{2}}{2} x$ .

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Step 8.4.2.1.2.1.1.1

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

$y = \frac{x^{2} x}{2} + \frac{x^{2}}{2} \cdot - 34 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.1.2

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

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Step 8.4.2.1.2.1.1.2.1

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

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Step 8.4.2.1.2.1.1.2.1.1

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

$y = \frac{x^{2} x^{1}}{2} + \frac{x^{2}}{2} \cdot - 34 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.1.2.1.2

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

$y = \frac{x^{2 + 1}}{2} + \frac{x^{2}}{2} \cdot - 34 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.1.2.2

Add $2$ and $1$ .

$y = \frac{x^{3}}{2} + \frac{x^{2}}{2} \cdot - 34 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.2

Cancel the common factor of $2$ .

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Step 8.4.2.1.2.1.2.1

Factor $2$ out of $- 34$ .

$y = \frac{x^{3}}{2} + \frac{x^{2}}{2} \cdot (2 (- 17)) - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.2.2

Cancel the common factor.

$y = \frac{x^{3}}{2} + \frac{x^{2}}{2} \cdot (2 \cdot - 17) - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.2.3

Rewrite the expression.

$y = \frac{x^{3}}{2} + x^{2} \cdot - 17 - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.3

Move $- 17$ to the left of $x^{2}$ .

$y = \frac{x^{3}}{2} - 17 \cdot x^{2} - 34 x \cdot x - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 8.4.2.1.2.1.4.1

Move $x$ .

$y = \frac{x^{3}}{2} - 17 x^{2} - 34 (x \cdot x) - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.4.2

Multiply $x$ by $x$ .

$y = \frac{x^{3}}{2} - 17 x^{2} - 34 x^{2} - 34 x \cdot - 34 + C x + C \cdot - 34$

Step 8.4.2.1.2.1.5

Multiply $- 34$ by $- 34$ .

$y = \frac{x^{3}}{2} - 17 x^{2} - 34 x^{2} + 1156 x + C x + C \cdot - 34$

Step 8.4.2.1.2.1.6

Move $- 34$ to the left of $C$ .

$y = \frac{x^{3}}{2} - 17 x^{2} - 34 x^{2} + 1156 x + C x - 34 C$

Step 8.4.2.1.2.2

Simplify by adding terms.

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Step 8.4.2.1.2.2.1

Subtract $34 x^{2}$ from $- 17 x^{2}$ .

$y = \frac{x^{3}}{2} - 51 x^{2} + 1156 x + C x - 34 C$

Step 8.4.2.1.2.2.2

Reorder.

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Step 8.4.2.1.2.2.2.1

Move $1156 x$ .

$y = \frac{x^{3}}{2} - 51 x^{2} + C x - 34 C + 1156 x$

Step 8.4.2.1.2.2.2.2

Move $- 51 x^{2}$ .

$y = \frac{x^{3}}{2} + C x - 51 x^{2} - 34 C + 1156 x$