Question

Express the solution of the following initial value problem in terms of an integral.

Answer 1

Answer:

$y = 2x + \frac{x^2}{2} - 6$

Step-by-step explanation:

Solution:-

- We are given the following initial value problem:

                             $\frac{dy}{dx} = 2 + x , y ( 2 ) = -6$

- We will first isolate the variables:

                            $dy = ( 2 + x ).dx$

- Perform integration for both sides of the equation:

                          $\int {dy} = \int {( 2 + x )}.dx + c\\\\y = 2x + \frac{x^2}{2} + c$

Where,

                         c: The constant of integration

- We will solve for the constant of integration by using the initial value y ( 0 ) = -6 as follows:

                         $-6 = 2(0) + \frac{0^2}{2} + c\\\\c = -6$

- The final solution can be expressed  as follows:

                         $y = 2x + \frac{x^2}{2} - 6$