Calculus Examples

$x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 0$

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 $x$ .

$x \frac{d v}{d x} + v = 0$

Step 2

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

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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 $g (x) = v$ .

$x \frac{d}{d x} [v] + v \frac{d}{d x} [x]$

Step 2.2

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

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

Step 2.3

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

$x v' + v \cdot 1$

Step 2.4

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

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

Step 2.5

Reorder $v$ and $1$ .

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

Step 2.6

Multiply $v$ by $1$ .

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

Step 3

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

$\frac{d}{d x} [x v] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x v] d x = \int 0 d x$

Step 5

Integrate the left side.

$x v = \int 0 d x$

Step 6

Integrate the right side.

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Step 6.1

The integral of $0$ with respect to $x$ is $0$ .

$x v = 0 + C_{1}$

Step 6.2

Add $0$ and $C_{1}$ .

$x v = C_{1}$

Step 7

Divide each term in $x v = C_{1}$ by $x$ and simplify.

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Step 7.1

Divide each term in $x v = C_{1}$ by $x$ .

$\frac{x v}{x} = \frac{C_{1}}{x}$

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.

$\frac{x v}{x} = \frac{C_{1}}{x}$

Step 7.2.1.2

Divide $v$ by $1$ .

$v = \frac{C_{1}}{x}$

Step 8

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

$\frac{d y}{d x} = \frac{C_{1}}{x}$

Step 9

Rewrite the equation.

$d y = \frac{C_{1}}{x} d x$

Step 10

Integrate both sides.

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Step 10.1

Set up an integral on each side.

$\int d y = \int \frac{C_{1}}{x} d x$

Step 10.2

Apply the constant rule.

$y + C_{2} = \int \frac{C_{1}}{x} d x$

Step 10.3

Integrate the right side.

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Step 10.3.1

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

$y + C_{2} = C_{1} \int \frac{1}{x} d x$

Step 10.3.2

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

$y + C_{2} = C_{1} (\ln (| x |) + C_{3})$

Step 10.3.3

Simplify.

$y + C_{2} = C_{1} \ln (| x |) + C_{4}$

Step 10.4

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

$y = C \ln (| x |) + D$