Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Factor $x$ out of $2 x y - x$ .

Tap for more steps...

Step 1.1.1

Factor $x$ out of $2 x y$ .

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

Step 1.1.2

Factor $x$ out of $- x$ .

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

Step 1.1.3

Factor $x$ out of $x (2 y) + x \cdot - 1$ .

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

Step 1.2

Multiply both sides by $\frac{1}{2 y - 1}$ .

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

Step 1.3

Cancel the common factor of $2 y - 1$ .

Tap for more steps...

Step 1.3.1

Factor $2 y - 1$ out of $x (2 y - 1)$ .

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u = 2 y - 1$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.1.1

Let $u = 2 y - 1$ . Find $\frac{d u}{d y}$ .

Tap for more steps...

Step 2.2.1.1.1

Differentiate $2 y - 1$ .

$\frac{d}{d y} [2 y - 1]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $2 y - 1$ with respect to $y$ is $\frac{d}{d y} [2 y] + \frac{d}{d y} [- 1]$ .

$\frac{d}{d y} [2 y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3

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

Tap for more steps...

Step 2.2.1.1.3.1

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

$2 \frac{d}{d y} [y] + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3.2

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

$2 \cdot 1 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d y} [- 1]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.2.1.1.4.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$2 + 0$

Step 2.2.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.1.2

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

$\int \frac{1}{u} \cdot \frac{1}{2} d u = \int x d x$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

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

$\int \frac{1}{u \cdot 2} d u = \int x d x$

Step 2.2.2.2

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

$\int \frac{1}{2 u} d u = \int x d x$

Step 2.2.3

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

$\frac{1}{2} \int \frac{1}{u} d u = \int x d x$

Step 2.2.4

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

$\frac{1}{2} (\ln (| u |) + C_{1}) = \int x d x$

Step 2.2.5

Simplify.

$\frac{1}{2} \ln (| u |) + C_{1} = \int x d x$

Step 2.2.6

Replace all occurrences of $u$ with $2 y - 1$ .

$\frac{1}{2} \ln (| 2 y - 1 |) + C_{1} = \int x d x$

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $2 (\frac{1}{2} \ln (| 2 y - 1 |))$ .

Tap for more steps...

Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $\ln (| 2 y - 1 |)$ .

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

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

Tap for more steps...

Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.2

Rewrite the expression.

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

Step 3.3

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Step 3.4

Rewrite $\ln (| 2 y - 1 |) = x^{2} + 2 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{x^{2} + 2 C} = | 2 y - 1 |$

Step 3.5

Solve for $y$ .

Tap for more steps...

Step 3.5.1

Rewrite the equation as $| 2 y - 1 | = e^{x^{2} + 2 C}$ .

$| 2 y - 1 | = e^{x^{2} + 2 C}$

Step 3.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 y - 1 = \pm e^{x^{2} + 2 C}$

Step 3.5.3

Add $1$ to both sides of the equation.

$2 y = \pm e^{x^{2} + 2 C} + 1$

Step 3.5.4

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

Tap for more steps...

Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

Tap for more steps...

Step 3.5.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

Tap for more steps...

Step 3.5.4.3.1

Combine the numerators over the common denominator.

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

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Simplify the constant of integration.

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

Step 4.2

Rewrite $e^{x^{2} + C}$ as $e^{x^{2}} e^{C}$ .

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

Step 4.3

Reorder $e^{x^{2}}$ and $e^{C}$ .

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

Step 4.4

Combine constants with the plus or minus.

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