Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Divide each term in $2 \frac{d y}{d x} + \frac{1}{x} y = x$ by $2$ .

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

Step 1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

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

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

Step 1.5

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

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

Step 1.6

Reorder terms.

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

Step 1.7

Factor $y$ out of $\frac{1}{2} \cdot \frac{y}{x}$ .

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

Step 1.8

Reorder $y$ and $\frac{1}{2} \cdot \frac{1}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{2} \cdot \frac{1}{x}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{1}{2} \cdot \frac{1}{x} d x}$

Step 2.2

Integrate $\frac{1}{2} \cdot \frac{1}{x}$ .

Tap for more steps...

Step 2.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{x}$ .

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

Step 2.2.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.2.3

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

$e^{\frac{1}{2} (\ln (| x |) + C)}$

Step 2.2.4

Simplify.

$e^{\frac{1}{2} \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\frac{1}{2} \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{\frac{1}{2}})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{\frac{1}{2}}$

Step 3

Multiply each term by the integrating factor $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.1

Multiply each term by $x^{\frac{1}{2}}$ .

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

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Combine.

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

Step 3.2.5

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 3.2.6

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 3.2.6.1

Move $x^{- \frac{1}{2}}$ .

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

Step 3.2.6.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

Tap for more steps...

Step 3.2.6.2.1

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

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

Step 3.2.6.2.2

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

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

Step 3.2.6.3

Write $1$ as a fraction with a common denominator.

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

Step 3.2.6.4

Combine the numerators over the common denominator.

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

Step 3.2.6.5

Add $- 1$ and $2$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.4.1

Move $x$ .

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

Step 3.4.2

Multiply $x$ by $x^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.3

Write $1$ as a fraction with a common denominator.

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

Step 3.4.4

Combine the numerators over the common denominator.

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

Step 3.4.5

Add $2$ and $1$ .

$x^{\frac{1}{2}} \frac{d y}{d x} + \frac{y}{2 x^{\frac{1}{2}}} = \frac{1}{2} x^{\frac{3}{2}}$

Step 3.5

Combine $\frac{1}{2}$ and $x^{\frac{3}{2}}$ .

$x^{\frac{1}{2}} \frac{d y}{d x} + \frac{y}{2 x^{\frac{1}{2}}} = \frac{x^{\frac{3}{2}}}{2}$

Step 4

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

$\frac{d}{d x} [x^{\frac{1}{2}} y] = \frac{x^{\frac{3}{2}}}{2}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{\frac{1}{2}} y] d x = \int \frac{x^{\frac{3}{2}}}{2} d x$

Step 6

Integrate the left side.

$x^{\frac{1}{2}} y = \int \frac{x^{\frac{3}{2}}}{2} d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$x^{\frac{1}{2}} y = \frac{1}{2} \int x^{\frac{3}{2}} d x$

Step 7.2

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$x^{\frac{1}{2}} y = \frac{1}{2} (\frac{2}{5} x^{\frac{5}{2}} + C)$

Step 7.3

Simplify the answer.

Tap for more steps...

Step 7.3.1

Rewrite $\frac{1}{2} (\frac{2}{5} x^{\frac{5}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{5} x^{\frac{5}{2}} + C$ .

$x^{\frac{1}{2}} y = \frac{1}{2} \cdot \frac{2}{5} x^{\frac{5}{2}} + C$

Step 7.3.2

Simplify.

Tap for more steps...

Step 7.3.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{5}$ .

$x^{\frac{1}{2}} y = \frac{2}{2 \cdot 5} x^{\frac{5}{2}} + C$

Step 7.3.2.2

Multiply $2$ by $5$ .

$x^{\frac{1}{2}} y = \frac{2}{10} x^{\frac{5}{2}} + C$

Step 7.3.2.3

Cancel the common factor of $2$ and $10$ .

Tap for more steps...

Step 7.3.2.3.1

Factor $2$ out of $2$ .

$x^{\frac{1}{2}} y = \frac{2 (1)}{10} x^{\frac{5}{2}} + C$

Step 7.3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 7.3.2.3.2.1

Factor $2$ out of $10$ .

$x^{\frac{1}{2}} y = \frac{2 \cdot 1}{2 \cdot 5} x^{\frac{5}{2}} + C$

Step 7.3.2.3.2.2

Cancel the common factor.

$x^{\frac{1}{2}} y = \frac{2 \cdot 1}{2 \cdot 5} x^{\frac{5}{2}} + C$

Step 7.3.2.3.2.3

Rewrite the expression.

$x^{\frac{1}{2}} y = \frac{1}{5} x^{\frac{5}{2}} + C$

Step 8

Divide each term in $x^{\frac{1}{2}} y = \frac{1}{5} x^{\frac{5}{2}} + C$ by $x^{\frac{1}{2}}$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $x^{\frac{1}{2}} y = \frac{1}{5} x^{\frac{5}{2}} + C$ by $x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} y}{x^{\frac{1}{2}}} = \frac{\frac{1}{5} x^{\frac{5}{2}}}{x^{\frac{1}{2}}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor.

$\frac{x^{\frac{1}{2}} y}{x^{\frac{1}{2}}} = \frac{\frac{1}{5} x^{\frac{5}{2}}}{x^{\frac{1}{2}}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.2.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{5} x^{\frac{5}{2}}}{x^{\frac{1}{2}}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.1

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$y = \frac{1}{5} x^{\frac{5}{2}} x^{- \frac{1}{2}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.2

Multiply $x^{\frac{5}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 8.3.1.2.1

Move $x^{- \frac{1}{2}}$ .

$y = \frac{1}{5} (x^{- \frac{1}{2}} x^{\frac{5}{2}}) + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.2.2

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

$y = \frac{1}{5} x^{- \frac{1}{2} + \frac{5}{2}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.2.3

Combine the numerators over the common denominator.

$y = \frac{1}{5} x^{\frac{- 1 + 5}{2}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.2.4

Add $- 1$ and $5$ .

$y = \frac{1}{5} x^{\frac{4}{2}} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.2.5

Divide $4$ by $2$ .

$y = \frac{1}{5} x^{2} + \frac{C}{x^{\frac{1}{2}}}$

Step 8.3.1.3

Combine $\frac{1}{5}$ and $x^{2}$ .

$y = \frac{x^{2}}{5} + \frac{C}{x^{\frac{1}{2}}}$