Question

Solve the differential equation and find the specific solution with the initial equation


`(dy)/(dx) =(1)/(2) y` and y(0) = 3

Answer 1

The solution of given differential equation
`y = 3 e^{(1)/(2)x }`

Explanation:

Step1:-

Given differential equation

`(dy)/(dx) = (1)/(2)y`

The differential operator form
`(D - (1)/(2))y =0`

The auxiliary equation is(A.E) f(m) = m - 1/2 =0

m = 1/2

The complementary solution is
`y(x) = c_(1) e^{a_(1) x} + c_(2) e^{a_(2) x}`

The complementary solution is
`y(x) = c_(1) e^{(1)/(2)x }` .......(1)

Step 2:-

Given conditions are x =0 and y(0) =3

From (1) we get

`y(0) = c_(1) e^{(1)/(2)0 }`

`3 = c_(1)`

now the solution of the given differential equation

substitute
`3 = c_(1)` in equation(1) , we get

`y(x) = 3 e^{(1)/(2)x }`

Final answer :-

The solution of given differential equation
`y = 3 e^{(1)/(2)x }`