Calculus Examples

$\frac{d y}{d x} = 5 x^{4} y - 4 x^{3} y$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Factor $x^{3} y$ out of $5 x^{4} y - 4 x^{3} y$ .

Tap for more steps...

Step 1.1.1

Factor $x^{3} y$ out of $5 x^{4} y$ .

$\frac{d y}{d x} = x^{3} y (5 x) - 4 x^{3} y$

Step 1.1.2

Factor $x^{3} y$ out of $- 4 x^{3} y$ .

$\frac{d y}{d x} = x^{3} y (5 x) + x^{3} y (- 4)$

Step 1.1.3

Factor $x^{3} y$ out of $x^{3} y (5 x) + x^{3} y (- 4)$ .

$\frac{d y}{d x} = x^{3} y (5 x - 4)$

Step 1.2

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

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (x^{3} y (5 x - 4))$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.3.1.1

Factor $y$ out of $x^{3} y (5 x - 4)$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y (x^{3} (5 x - 4)))$

Step 1.3.1.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (y (x^{3} (5 x - 4)))$

Step 1.3.1.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = x^{3} (5 x - 4)$

Step 1.3.2

Apply the distributive property.

$\frac{1}{y} \frac{d y}{d x} = x^{3} (5 x) + x^{3} \cdot - 4$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$\frac{1}{y} \frac{d y}{d x} = 5 x^{3} x + x^{3} \cdot - 4$

Step 1.3.4

Move $- 4$ to the left of $x^{3}$ .

$\frac{1}{y} \frac{d y}{d x} = 5 x^{3} x - 4 \cdot x^{3}$

Step 1.3.5

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

Tap for more steps...

Step 1.3.5.1

Move $x$ .

$\frac{1}{y} \frac{d y}{d x} = 5 (x \cdot x^{3}) - 4 \cdot x^{3}$

Step 1.3.5.2

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

Tap for more steps...

Step 1.3.5.2.1

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

$\frac{1}{y} \frac{d y}{d x} = 5 (x^{1} x^{3}) - 4 \cdot x^{3}$

Step 1.3.5.2.2

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

$\frac{1}{y} \frac{d y}{d x} = 5 x^{1 + 3} - 4 \cdot x^{3}$

Step 1.3.5.3

Add $1$ and $3$ .

$\frac{1}{y} \frac{d y}{d x} = 5 x^{4} - 4 \cdot x^{3}$

$\frac{1}{y} \frac{d y}{d x} = 5 x^{4} - 4 x^{3}$

Step 1.4

Rewrite the equation.

$\frac{1}{y} d y = (5 x^{4} - 4 x^{3}) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 5 x^{4} - 4 x^{3} d x$

Step 2.2

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

$\ln (| y |) + C_{1} = \int 5 x^{4} - 4 x^{3} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$\ln (| y |) + C_{1} = \int 5 x^{4} d x + \int - 4 x^{3} d x$

Step 2.3.2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\ln (| y |) + C_{1} = 5 \int x^{4} d x + \int - 4 x^{3} d x$

Step 2.3.3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\ln (| y |) + C_{1} = 5 (\frac{1}{5} x^{5} + C_{2}) + \int - 4 x^{3} d x$

Step 2.3.4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$\ln (| y |) + C_{1} = 5 (\frac{1}{5} x^{5} + C_{2}) - 4 \int x^{3} d x$

Step 2.3.5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\ln (| y |) + C_{1} = 5 (\frac{1}{5} x^{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3})$

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

Simplify.

$\ln (| y |) + C_{1} = 5 (\frac{1}{5}) x^{5} - 4 (\frac{1}{4}) x^{4} + C_{4}$

Step 2.3.6.2

Simplify.

Tap for more steps...

Step 2.3.6.2.1

Combine $5$ and $\frac{1}{5}$ .

$\ln (| y |) + C_{1} = \frac{5}{5} x^{5} - 4 (\frac{1}{4}) x^{4} + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.3.6.2.2.1

Cancel the common factor.

$\ln (| y |) + C_{1} = \frac{5}{5} x^{5} - 4 (\frac{1}{4}) x^{4} + C_{4}$

Step 2.3.6.2.2.2

Rewrite the expression.

$\ln (| y |) + C_{1} = 1 x^{5} - 4 (\frac{1}{4}) x^{4} + C_{4}$

Step 2.3.6.2.3

Multiply $x^{5}$ by $1$ .

$\ln (| y |) + C_{1} = x^{5} - 4 (\frac{1}{4}) x^{4} + C_{4}$

Step 2.3.6.2.4

Combine $- 4$ and $\frac{1}{4}$ .

$\ln (| y |) + C_{1} = x^{5} + \frac{- 4}{4} x^{4} + C_{4}$

Step 2.3.6.2.5

Cancel the common factor of $- 4$ and $4$ .

Tap for more steps...

Step 2.3.6.2.5.1

Factor $4$ out of $- 4$ .

$\ln (| y |) + C_{1} = x^{5} + \frac{4 \cdot - 1}{4} x^{4} + C_{4}$

Step 2.3.6.2.5.2

Cancel the common factors.

Tap for more steps...

Step 2.3.6.2.5.2.1

Factor $4$ out of $4$ .

$\ln (| y |) + C_{1} = x^{5} + \frac{4 \cdot - 1}{4 (1)} x^{4} + C_{4}$

Step 2.3.6.2.5.2.2

Cancel the common factor.

$\ln (| y |) + C_{1} = x^{5} + \frac{4 \cdot - 1}{4 \cdot 1} x^{4} + C_{4}$

Step 2.3.6.2.5.2.3

Rewrite the expression.

$\ln (| y |) + C_{1} = x^{5} + \frac{- 1}{1} x^{4} + C_{4}$

Step 2.3.6.2.5.2.4

Divide $- 1$ by $1$ .

$\ln (| y |) + C_{1} = x^{5} - x^{4} + C_{4}$

Step 2.4

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

$\ln (| y |) = x^{5} - x^{4} + C$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

$e^{\ln (| y |)} = e^{x^{5} - x^{4} + C}$

Step 3.2

Rewrite $\ln (| y |) = x^{5} - x^{4} + C$ 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 $b^{y} = x$ .

$e^{x^{5} - x^{4} + C} = | y |$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $| y | = e^{x^{5} - x^{4} + C}$ .

$| y | = e^{x^{5} - x^{4} + C}$

Step 3.3.2

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

$y = \pm e^{x^{5} - x^{4} + C}$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Rewrite $e^{x^{5} - x^{4} + C}$ as $e^{x^{5} - x^{4}} e^{C}$ .

$y = \pm e^{x^{5} - x^{4}} e^{C}$

Step 4.2

Reorder $e^{x^{5} - x^{4}}$ and $e^{C}$ .

$y = \pm e^{C} e^{x^{5} - x^{4}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{x^{5} - x^{4}}$