Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Factor $y$ out of $x^{2} y$ .

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

Step 1.1.2

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

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

Step 1.1.3

Factor $y$ out of $3 y$ .

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

Step 1.1.4

Factor $y$ out of $y x^{2} + y (- x)$ .

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

Step 1.1.5

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

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

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

$\ln (| y |) + C_{1} = \int x^{2} - x + 3 d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int x^{2} d x + \int - x d x + \int 3 d x$

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

$\ln (| y |) + C_{1} = \frac{1}{3} x^{3} + C_{2} - (\frac{1}{2} x^{2} + C_{3}) + 3 x + C_{4}$

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

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

$\ln (| y |) + C_{1} = \frac{1}{3} x^{3} + C_{2} - (\frac{x^{2}}{2} + C_{3}) + 3 x + C_{4}$

Step 2.3.6.2

Simplify.

$\ln (| y |) + C_{1} = \frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 3 x + C_{5}$

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$e^{\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 3 x + C} = | y |$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $| y | = e^{\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 3 x + C}$ .

$| y | = e^{\frac{1}{3} x^{3} - \frac{1}{2} x^{2} + 3 x + C}$

Step 3.3.2

Simplify each term.

Tap for more steps...

Step 3.3.2.1

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

$| y | = e^{\frac{x^{3}}{3} - \frac{1}{2} x^{2} + 3 x + C}$

Step 3.3.2.2

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

$| y | = e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x + C}$

Step 3.3.3

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

$y = \pm e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x + C}$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Rewrite $e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x + C}$ as $e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x} e^{C}$ .

$y = \pm e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x} e^{C}$

Step 4.2

Reorder $e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{x^{3}}{3} - \frac{x^{2}}{2} + 3 x}$