Theoretical question on solutions

[nhrock3]

we have y'=f(x)y+g(x) when f and g are continues on (a,b).
and there is a point x_0 for which g(x_0) differs zero.suppose that u_1 and u_2 are
solution to the equation on (a,b) .and if for some c,d d*u_1+c*u_2 is also a solution
thn we cn conclude that : (we need to choose one of the options)
A. c or d is zero
B. c=1-d
C. c^2+d^2=1
D.if u_1 and u_2 are
solution to the equation then d*u_1+c*u_2 is also a solution


i only know that on (a,b) there could be a single solution for the initial condition.
thats my theoretical knowledge on the subject

 

[Mark44]

we have y'=f(x)y+g(x) when f and g are continues on (a,b).
and there is a point x_0 for which g(x_0) differs zero.suppose that u_1 and u_2 are
solution to the equation on (a,b) .and if for some c,d d*u_1+c*u_2 is also a solution
thn we cn conclude that : (we need to choose one of the options)
A. c or d is zero
B. c=1-d
C. c^2+d^2=1
D.if u_1 and u_2 are
solution to the equation then d*u_1+c*u_2 is also a solution


i only know that on (a,b) there could be a single solution for the initial condition.
thats my theoretical knowledge on the subject

Start working with your given information. For example, since u₁ and u₂ are solutions to the diff. equation, then
u₁'(x) = f(x)u₁(x) + g(x), and
u₂'(x) = f(x)u₂(x) + g(x).

It also says that for some c and d, cu₁ + du₂ is a solution. What does that mean in terms of the given differential equation?