Calculus Examples

$\frac{d p}{d z} + \frac{2}{z} p = 4$

Step 1

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

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Step 1.1

Set up the integration.

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$e^{2 (\ln (| z |) + C)}$

Step 1.2.3

Simplify.

$e^{2 \ln (| z |) + C}$

Step 1.3

Remove the constant of integration.

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

Step 1.4

Use the logarithmic power rule.

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

Step 1.5

Exponentiation and log are inverse functions.

$z^{2}$

Step 2

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

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Step 2.1

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

$z^{2} \frac{d p}{d z} + z^{2} (\frac{2}{z} p) = z^{2} \cdot 4$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{2}{z}$ and $p$ .

$z^{2} \frac{d p}{d z} + z^{2} \frac{2 p}{z} = z^{2} \cdot 4$

Step 2.2.2

Cancel the common factor of $z$ .

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Step 2.2.2.1

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

$z^{2} \frac{d p}{d z} + z \cdot z \frac{2 p}{z} = z^{2} \cdot 4$

Step 2.2.2.2

Cancel the common factor.

$z^{2} \frac{d p}{d z} + z \cdot z \frac{2 p}{z} = z^{2} \cdot 4$

Step 2.2.2.3

Rewrite the expression.

$z^{2} \frac{d p}{d z} + z (2 p) = z^{2} \cdot 4$

Step 2.2.3

Rewrite using the commutative property of multiplication.

$z^{2} \frac{d p}{d z} + 2 z p = z^{2} \cdot 4$

Step 2.3

Move $4$ to the left of $z^{2}$ .

$z^{2} \frac{d p}{d z} + 2 z p = 4 z^{2}$

Step 3

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

$\frac{d}{d z} [z^{2} p] = 4 z^{2}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d z} [z^{2} p] d z = \int 4 z^{2} d z$

Step 5

Integrate the left side.

$z^{2} p = \int 4 z^{2} d z$

Step 6

Integrate the right side.

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Step 6.1

Since $4$ is constant with respect to $z$ , move $4$ out of the integral.

$z^{2} p = 4 \int z^{2} d z$

Step 6.2

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

$z^{2} p = 4 (\frac{1}{3} z^{3} + C)$

Step 6.3

Simplify the answer.

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Step 6.3.1

Rewrite $4 (\frac{1}{3} z^{3} + C)$ as $4 (\frac{1}{3}) z^{3} + C$ .

$z^{2} p = 4 (\frac{1}{3}) z^{3} + C$

Step 6.3.2

Combine $4$ and $\frac{1}{3}$ .

$z^{2} p = \frac{4}{3} z^{3} + C$

Step 7

Divide each term in $z^{2} p = \frac{4}{3} z^{3} + C$ by $z^{2}$ and simplify.

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Step 7.1

Divide each term in $z^{2} p = \frac{4}{3} z^{3} + C$ by $z^{2}$ .

$\frac{z^{2} p}{z^{2}} = \frac{\frac{4}{3} z^{3}}{z^{2}} + \frac{C}{z^{2}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{z^{2} p}{z^{2}} = \frac{\frac{4}{3} z^{3}}{z^{2}} + \frac{C}{z^{2}}$

Step 7.2.1.2

Divide $p$ by $1$ .

$p = \frac{\frac{4}{3} z^{3}}{z^{2}} + \frac{C}{z^{2}}$

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 $z^{3}$ and $z^{2}$ .

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Step 7.3.1.1.1

Factor $z^{2}$ out of $\frac{4}{3} z^{3}$ .

$p = \frac{z^{2} (\frac{4}{3} z)}{z^{2}} + \frac{C}{z^{2}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$p = \frac{z^{2} (\frac{4}{3} z)}{z^{2} \cdot 1} + \frac{C}{z^{2}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$p = \frac{z^{2} (\frac{4}{3} z)}{z^{2} \cdot 1} + \frac{C}{z^{2}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$p = \frac{\frac{4}{3} z}{1} + \frac{C}{z^{2}}$

Step 7.3.1.1.2.4

Divide $\frac{4}{3} z$ by $1$ .

$p = \frac{4}{3} z + \frac{C}{z^{2}}$

Step 7.3.1.2

Combine $\frac{4}{3}$ and $z$ .

$p = \frac{4 z}{3} + \frac{C}{z^{2}}$