Calculus Examples

$x^{2} \frac{d y}{d x} + 2 x y = \ln (x)$ , $y (1) = 2$

Step 1

Check if the left side of the equation is the result of the derivative of the term $x^{2} y$ .

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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 $f (x) = x^{2}$ and $g (x) = y$ .

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

Step 1.2

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

$x^{2} y' + y \frac{d}{d x} [x^{2}]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} y' + y (2 x)$

Step 1.4

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

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

Step 1.5

Remove parentheses.

$x^{2} \frac{d y}{d x} + y \cdot 2 x$

Step 1.6

Move $y$ .

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

Step 2

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

$\frac{d}{d x} [x^{2} y] = \ln (x)$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int \ln (x) d x$

Step 4

Integrate the left side.

$x^{2} y = \int \ln (x) d x$

Step 5

Integrate the right side.

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Step 5.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Rewrite the expression.

$x^{2} y = \ln (x) x - \int d x$

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $y$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

Factor $x$ out of $\ln (x) x$ .

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

Step 6.3.1.1.2

Cancel the common factors.

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Step 6.3.1.1.2.1

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

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

Step 6.3.1.1.2.2

Cancel the common factor.

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

Step 6.3.1.1.2.3

Rewrite the expression.

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

Step 6.3.1.2

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

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Step 6.3.1.2.1

Factor $x$ out of $- x$ .

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

Step 6.3.1.2.2

Cancel the common factors.

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Step 6.3.1.2.2.1

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

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

Step 6.3.1.2.2.2

Cancel the common factor.

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

Step 6.3.1.2.2.3

Rewrite the expression.

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

Step 6.3.1.3

Move the negative in front of the fraction.

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

Step 7

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $2$ for $y$ in $y = \frac{\ln (x)}{x} - \frac{1}{x} + \frac{C}{x^{2}}$ .

$2 = \frac{\ln (1)}{1} - \frac{1}{1} + \frac{C}{1^{2}}$

Step 8

Solve for $C$ .

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Step 8.1

Rewrite the equation as $\frac{\ln (1)}{1} - \frac{1}{1} + \frac{C}{1^{2}} = 2$ .

$\frac{\ln (1)}{1} - \frac{1}{1} + \frac{C}{1^{2}} = 2$

Step 8.2

Simplify $\frac{\ln (1)}{1} - \frac{1}{1} + \frac{C}{1^{2}}$ .

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Step 8.2.1

Combine the numerators over the common denominator.

$\frac{\ln (1) - 1}{1} + \frac{C}{1^{2}} = 2$

Step 8.2.2

The natural logarithm of $1$ is $0$ .

$\frac{0 - 1}{1} + \frac{C}{1^{2}} = 2$

Step 8.2.3

Subtract $1$ from $0$ .

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

Step 8.2.4

Simplify each term.

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Step 8.2.4.1

Divide $- 1$ by $1$ .

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

Step 8.2.4.2

One to any power is one.

$- 1 + \frac{C}{1} = 2$

Step 8.2.4.3

Divide $C$ by $1$ .

$- 1 + C = 2$

Step 8.3

Move all terms not containing $C$ to the right side of the equation.

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Step 8.3.1

Add $1$ to both sides of the equation.

$C = 2 + 1$

Step 8.3.2

Add $2$ and $1$ .

$C = 3$

Step 9

Substitute $3$ for $C$ in $y = \frac{\ln (x)}{x} - \frac{1}{x} + \frac{C}{x^{2}}$ and simplify.

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Step 9.1

Substitute $3$ for $C$ .

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