Calculus Examples

$\frac{d y}{d x} - 4 = 8 x^{3} - 3 x^{2}$ , $y (1) = 5$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Add $4$ to both sides of the equation.

$\frac{d y}{d x} = 8 x^{3} - 3 x^{2} + 4$

Step 1.2

Rewrite the equation.

$d y = (8 x^{3} - 3 x^{2} + 4) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 8 x^{3} - 3 x^{2} + 4 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 x^{3} - 3 x^{2} + 4 d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 8 x^{3} d x + \int - 3 x^{2} d x + \int 4 d x$

Step 2.3.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$y + C_{1} = 8 \int x^{3} d x + \int - 3 x^{2} d x + \int 4 d x$

Step 2.3.3

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

$y + C_{1} = 8 (\frac{1}{4} x^{4} + C_{2}) + \int - 3 x^{2} d x + \int 4 d x$

Step 2.3.4

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$y + C_{1} = 8 (\frac{1}{4} x^{4} + C_{2}) - 3 \int x^{2} d x + \int 4 d x$

Step 2.3.5

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

$y + C_{1} = 8 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + \int 4 d x$

Step 2.3.6

Apply the constant rule.

$y + C_{1} = 8 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + 4 x + C_{4}$

Step 2.3.7

Simplify.

Tap for more steps...

Step 2.3.7.1

Simplify.

Tap for more steps...

Step 2.3.7.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y + C_{1} = 8 (\frac{x^{4}}{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + 4 x + C_{4}$

Step 2.3.7.1.2

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

$y + C_{1} = 8 (\frac{x^{4}}{4} + C_{2}) - 3 (\frac{x^{3}}{3} + C_{3}) + 4 x + C_{4}$

Step 2.3.7.2

Simplify.

$y + C_{1} = 2 x^{4} - x^{3} + 4 x + C_{5}$

Step 2.4

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

$y = 2 x^{4} - x^{3} + 4 x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $5$ for $y$ in $y = 2 x^{4} - x^{3} + 4 x + K$ .

$5 = 2 \cdot 1^{4} - 1^{3} + 4 \cdot 1 + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $2 \cdot 1^{4} - 1^{3} + 4 \cdot 1 + K = 5$ .

$2 \cdot 1^{4} - 1^{3} + 4 \cdot 1 + K = 5$

Step 4.2

Simplify $2 \cdot 1^{4} - 1^{3} + 4 \cdot 1 + K$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

One to any power is one.

$2 \cdot 1 - 1^{3} + 4 \cdot 1 + K = 5$

Step 4.2.1.2

Multiply $2$ by $1$ .

$2 - 1^{3} + 4 \cdot 1 + K = 5$

Step 4.2.1.3

One to any power is one.

$2 - 1 \cdot 1 + 4 \cdot 1 + K = 5$

Step 4.2.1.4

Multiply $- 1$ by $1$ .

$2 - 1 + 4 \cdot 1 + K = 5$

Step 4.2.1.5

Multiply $4$ by $1$ .

$2 - 1 + 4 + K = 5$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Subtract $1$ from $2$ .

$1 + 4 + K = 5$

Step 4.2.2.2

Add $1$ and $4$ .

$5 + K = 5$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

Tap for more steps...

Step 4.3.1

Subtract $5$ from both sides of the equation.

$K = 5 - 5$

Step 4.3.2

Subtract $5$ from $5$ .

$K = 0$

Step 5

Substitute $0$ for $K$ in $y = 2 x^{4} - x^{3} + 4 x + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $0$ for $K$ .

$y = 2 x^{4} - x^{3} + 4 x + 0$

Step 5.2

Add $2 x^{4} - x^{3} + 4 x$ and $0$ .

$y = 2 x^{4} - x^{3} + 4 x$