Question

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

dy/dx = 2 + x, y(2 ) = -6

Answer 1

`y = 2x + (x^2)/(2) - 6`

Explanation:

Solution:-

- We are given the following initial value problem:

`(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 + (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) + (0^2)/(2) + c\\\\c = -6`

- The final solution can be expressed as follows:

`y = 2x + (x^2)/(2) - 6`