Calculus Examples

$\frac{d y}{d x} = 10 - x$ , $y (0) = - 1$

Step 1

Rewrite the equation.

$d y = (10 - x) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 10 - x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 10 - x d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 10 d x + \int - x d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = 10 x + C_{2} + \int - x d x$

Step 2.3.3

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

$y + C_{1} = 10 x + C_{2} - \int x d x$

Step 2.3.4

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

$y + C_{1} = 10 x + C_{2} - (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.5

Simplify.

$y + C_{1} = 10 x - \frac{1}{2} x^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$y + C_{1} = - \frac{1}{2} x^{2} + 10 x + C_{4}$

Step 2.4

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

$y = - \frac{1}{2} x^{2} + 10 x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $- 1$ for $y$ in $y = - \frac{1}{2} x^{2} + 10 x + K$ .

$- 1 = - \frac{1}{2} \cdot 0^{2} + 10 \cdot 0 + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $- \frac{1}{2} \cdot 0^{2} + 10 \cdot 0 + K = - 1$ .

$- \frac{1}{2} \cdot 0^{2} + 10 \cdot 0 + K = - 1$

Step 4.2

Simplify $- \frac{1}{2} \cdot 0^{2} + 10 \cdot 0 + K$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{2} \cdot 0 + 10 \cdot 0 + K = - 1$

Step 4.2.1.2

Multiply $- \frac{1}{2} \cdot 0$ .

Tap for more steps...

Step 4.2.1.2.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{2}) + 10 \cdot 0 + K = - 1$

Step 4.2.1.2.2

Multiply $0$ by $\frac{1}{2}$ .

$0 + 10 \cdot 0 + K = - 1$

Step 4.2.1.3

Multiply $10$ by $0$ .

$0 + 0 + K = - 1$

Step 4.2.2

Combine the opposite terms in $0 + 0 + K$ .

Tap for more steps...

Step 4.2.2.1

Add $0$ and $0$ .

$0 + K = - 1$

Step 4.2.2.2

Add $0$ and $K$ .

$K = - 1$

Step 5

Substitute $- 1$ for $K$ in $y = - \frac{1}{2} x^{2} + 10 x + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $- 1$ for $K$ .

$y = - \frac{1}{2} x^{2} + 10 x - 1$

Step 5.2

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

$y = - \frac{x^{2}}{2} + 10 x - 1$