Calculus Examples

$\frac{d y}{d x} - \frac{3 y}{x} = x^{3}$ if $y (1) = 4$

Step 1

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

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Step 1.1

Factor $y$ out of $- \frac{3 y}{x}$ .

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

Step 1.2

Reorder $y$ and $- \frac{3}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{3}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int - \frac{3}{x} d x}$

Step 2.2

Integrate $- \frac{3}{x}$ .

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Step 2.2.1

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

$e^{- \int \frac{3}{x} d x}$

Step 2.2.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{- (3 \int \frac{1}{x} d x)}$

Step 2.2.3

Multiply $3$ by $- 1$ .

$e^{- 3 \int \frac{1}{x} d x}$

Step 2.2.4

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

$e^{- 3 (\ln (| x |) + C)}$

Step 2.2.5

Simplify.

$e^{- 3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 3}$

Step 2.6

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

$\frac{1}{x^{3}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{3}}$ .

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Step 3.1

Multiply each term by $\frac{1}{x^{3}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x^{3}}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{1}{x^{3}} (\frac{3}{x} y) = \frac{1}{x^{3}} x^{3}$

Step 3.2.3

Combine $\frac{3}{x}$ and $y$ .

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{1}{x^{3}} \cdot \frac{3 y}{x} = \frac{1}{x^{3}} x^{3}$

Step 3.2.4

Multiply $- \frac{1}{x^{3}} \cdot \frac{3 y}{x}$ .

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Step 3.2.4.1

Multiply $\frac{3 y}{x}$ by $\frac{1}{x^{3}}$ .

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x \cdot x^{3}} = \frac{1}{x^{3}} x^{3}$

Step 3.2.4.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.2.4.2.1

Multiply $x$ by $x^{3}$ .

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Step 3.2.4.2.1.1

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

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x^{1} x^{3}} = \frac{1}{x^{3}} x^{3}$

Step 3.2.4.2.1.2

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

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x^{1 + 3}} = \frac{1}{x^{3}} x^{3}$

Step 3.2.4.2.2

Add $1$ and $3$ .

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x^{4}} = \frac{1}{x^{3}} x^{3}$

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x^{4}} = \frac{1}{x^{3}} x^{3}$

Step 3.3.2

Rewrite the expression.

$\frac{\frac{d y}{d x}}{x^{3}} - \frac{3 y}{x^{4}} = 1$

Step 4

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

$\frac{d}{d x} [\frac{1}{x^{3}} y] = 1$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{3}} y] d x = \int d x$

Step 6

Integrate the left side.

$\frac{1}{x^{3}} y = \int d x$

Step 7

Apply the constant rule.

$\frac{1}{x^{3}} y = x + C$

Step 8

Solve for $y$ .

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Step 8.1

Combine $\frac{1}{x^{3}}$ and $y$ .

$\frac{y}{x^{3}} = x + C$

Step 8.2

Multiply both sides by $x^{3}$ .

$\frac{y}{x^{3}} x^{3} = (x + C) x^{3}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

Cancel the common factor.

$\frac{y}{x^{3}} x^{3} = (x + C) x^{3}$

Step 8.3.1.1.2

Rewrite the expression.

$y = (x + C) x^{3}$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(x + C) x^{3}$ .

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Step 8.3.2.1.1

Apply the distributive property.

$y = x \cdot x^{3} + C x^{3}$

Step 8.3.2.1.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 8.3.2.1.2.1

Multiply $x$ by $x^{3}$ .

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Step 8.3.2.1.2.1.1

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

$y = x^{1} x^{3} + C x^{3}$

Step 8.3.2.1.2.1.2

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

$y = x^{1 + 3} + C x^{3}$

Step 8.3.2.1.2.2

Add $1$ and $3$ .

$y = x^{4} + C x^{3}$

Step 8.3.2.1.3

Reorder $x^{4}$ and $C x^{3}$ .

$y = C x^{3} + x^{4}$

Step 9

Use the initial condition to find the value of $C$ by substituting $1$ for $x$ and $4$ for $y$ in $y = C x^{3} + x^{4}$ .

$4 = C \cdot 1^{3} + 1^{4}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $C \cdot 1^{3} + 1^{4} = 4$ .

$C \cdot 1^{3} + 1^{4} = 4$

Step 10.2

Simplify each term.

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Step 10.2.1

One to any power is one.

$C \cdot 1 + 1^{4} = 4$

Step 10.2.2

Multiply $C$ by $1$ .

$C + 1^{4} = 4$

Step 10.2.3

One to any power is one.

$C + 1 = 4$

Step 10.3

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

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Step 10.3.1

Subtract $1$ from both sides of the equation.

$C = 4 - 1$

Step 10.3.2

Subtract $1$ from $4$ .

$C = 3$

Step 11

Substitute $3$ for $C$ in $y = C x^{3} + x^{4}$ and simplify.

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Step 11.1

Substitute $3$ for $C$ .

$y = 3 x^{3} + x^{4}$