Question

How do you solve the initial value problem?

(dy/dx)-2y+4=0, y(1)=4

Answer 1

I am going to use t instead of x. I can do initial condition better that way
let y = a e^kt + b
dy/dt = a k e^kt

a k e^kt -2 a e^kt -2b + 4 = 0

so b = 2
a k = 2 a
k = 2

y = a e^2t + 2
when t = 1, y = 4
4 = a e^2 +2
a e^2 = 2
a = 2/e^2
so
y = (2/e^2) e^2t + 2
y = 2 e^(2t-2) + 2