Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Subtract $y$ from both sides of the equation.

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

Step 1.2

Divide each term in $(2 x + 3) \frac{d y}{d x} - y = {(2 x + 3)}^{\frac{1}{2}}$ by $2 x + 3$ .

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

Step 1.3

Cancel the common factor of $2 x + 3$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

Factor $2 x + 3$ out of ${(2 x + 3)}^{\frac{1}{2}}$ .

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

Step 1.5

Cancel the common factors.

Tap for more steps...

Step 1.5.1

Multiply by $1$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

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

Step 1.5.4

Divide ${(2 x + 3)}^{- \frac{1}{2}}$ by $1$ .

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

Step 1.6

Factor $y$ out of $\frac{- y}{2 x + 3}$ .

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

Step 1.7

Reorder $y$ and $\frac{- 1}{2 x + 3}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{2 x + 3}$ .

Tap for more steps...

Step 2.1

Set up the integration.

$e^{\int \frac{- 1}{2 x + 3} d x}$

Step 2.2

Integrate $\frac{- 1}{2 x + 3}$ .

Tap for more steps...

Step 2.2.1

Move the negative in front of the fraction.

$e^{\int - \frac{1}{2 x + 3} d x}$

Step 2.2.2

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

$e^{- \int \frac{1}{2 x + 3} d x}$

Step 2.2.3

Let $u = 2 x + 3$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.3.1

Let $u = 2 x + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.2.3.1.1

Differentiate $2 x + 3$ .

$\frac{d}{d x} [2 x + 3]$

Step 2.2.3.1.2

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$

Step 2.2.3.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

Tap for more steps...

Step 2.2.3.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.3.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 2.2.3.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [3]$

Step 2.2.3.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.2.3.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.2.3.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.3.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.2.4

Simplify.

Tap for more steps...

Step 2.2.4.1

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

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

Step 2.2.4.2

Move $2$ to the left of $u$ .

$e^{- \int \frac{1}{2 u} d u}$

Step 2.2.5

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

$e^{- (\frac{1}{2} \int \frac{1}{u} d u)}$

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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

Step 2.2.8

Replace all occurrences of $u$ with $2 x + 3$ .

$e^{- \frac{1}{2} \ln (| 2 x + 3 |) + C}$

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

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

Step 2.6

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

$\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$ .

Tap for more steps...

Step 3.1

Multiply each term by $\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$ .

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

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Combine $\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $- \frac{1}{{(2 x + 3)}^{\frac{1}{2}}} \cdot \frac{y}{2 x + 3}$ .

Tap for more steps...

Step 3.2.5.1

Multiply $\frac{y}{2 x + 3}$ by $\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$ .

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

Step 3.2.5.2

Multiply $2 x + 3$ by ${(2 x + 3)}^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 3.2.5.2.1

Multiply $2 x + 3$ by ${(2 x + 3)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 3.2.5.2.1.1

Raise $2 x + 3$ to the power of $1$ .

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

Step 3.2.5.2.1.2

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

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

Step 3.2.5.2.2

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

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

Step 3.2.5.2.3

Combine the numerators over the common denominator.

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

Step 3.2.5.2.4

Add $2$ and $1$ .

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

Step 3.3

Cancel the common factor of ${(2 x + 3)}^{- \frac{1}{2}}$ .

Tap for more steps...

Step 3.3.1

Factor ${(2 x + 3)}^{- \frac{1}{2}}$ out of ${(2 x + 3)}^{\frac{1}{2}}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

Simplify.

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

Step 7.2

Let $u = 2 x + 3$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 7.2.1

Let $u = 2 x + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 7.2.1.1

Differentiate $2 x + 3$ .

$\frac{d}{d x} [2 x + 3]$

Step 7.2.1.2

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$

Step 7.2.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

Tap for more steps...

Step 7.2.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 7.2.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 7.2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [3]$

Step 7.2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 7.2.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 + 0$

Step 7.2.1.4.2

Add $2$ and $0$ .

$2$

Step 7.2.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{{(2 x + 3)}^{\frac{1}{2}}} y = \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 7.3

Simplify.

Tap for more steps...

Step 7.3.1

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

$\frac{1}{{(2 x + 3)}^{\frac{1}{2}}} y = \int \frac{1}{u \cdot 2} d u$

Step 7.3.2

Move $2$ to the left of $u$ .

$\frac{1}{{(2 x + 3)}^{\frac{1}{2}}} y = \int \frac{1}{2 u} d u$

Step 7.4

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

$\frac{1}{{(2 x + 3)}^{\frac{1}{2}}} y = \frac{1}{2} \int \frac{1}{u} d u$

Step 7.5

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

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

Step 7.6

Simplify.

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

Step 7.7

Replace all occurrences of $u$ with $2 x + 3$ .

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

Step 8

Solve for $y$ .

Tap for more steps...

Step 8.1

Combine $\frac{1}{{(2 x + 3)}^{\frac{1}{2}}}$ and $y$ .

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

Step 8.2

Simplify $\frac{1}{2} \ln (| 2 x + 3 |)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 8.3

Multiply both sides by ${(2 x + 3)}^{\frac{1}{2}}$ .

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

Step 8.4

Simplify.

Tap for more steps...

Step 8.4.1

Simplify the left side.

Tap for more steps...

Step 8.4.1.1

Cancel the common factor of ${(2 x + 3)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 8.4.1.1.1

Cancel the common factor.

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

Step 8.4.1.1.2

Rewrite the expression.

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

Step 8.4.2

Simplify the right side.

Tap for more steps...

Step 8.4.2.1

Simplify $(\ln ({| 2 x + 3 |}^{\frac{1}{2}}) + C) {(2 x + 3)}^{\frac{1}{2}}$ .

Tap for more steps...

Step 8.4.2.1.1

Apply the distributive property.

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

Step 8.4.2.1.2

Simplify the expression.

Tap for more steps...

Step 8.4.2.1.2.1

Reorder factors in $\ln ({| 2 x + 3 |}^{\frac{1}{2}}) {(2 x + 3)}^{\frac{1}{2}} + C {(2 x + 3)}^{\frac{1}{2}}$ .

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

Step 8.4.2.1.2.2

Reorder ${(2 x + 3)}^{\frac{1}{2}} \ln ({| 2 x + 3 |}^{\frac{1}{2}})$ and $C {(2 x + 3)}^{\frac{1}{2}}$ .

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