Question about solving ODE with Complex eigenvalue

[nufeng]

For example,
ODE: y'' + y = 0
solve this problem using MAPLE
f(x) = _C1*sin(x)+_C2*cos(x)

My question is Eigenvalue for D^2+1=0 is +i, -i
so general solution is f(x) = C1*exp(i*x)+C2*exp(-i*x)
according to Euler's formula f(x) = C1( cos(x)+i*sin(x) ) + C2*( cos(x)-i*sin(x) )
it is different from the general solution generated by MAPLE
why?

Thanks!

 

[AlephZero]

Your solution is the general solution assuming f(x) is complex, and your constants C1 and C2 are also complex. You can rearrange it as
f(x) = (C1 + C2) cos(x) + i (C1 - C2) sin(x)
or
f(x) = A1 cos(x) + A2 sin(x)
wherne A1 and A2 are complex constants.
If course if you want to restrict f(x) to be a real function, A1 and A2 must be real. That condition is the equivalent to C1 and C2 being complex conjugates, so that C1 + C2 is real and C1 - C2 is imaginary.

 

[nufeng]

AlephZero, thank you! Really helpful!

Your solution is the general solution assuming f(x) is complex, and your constants C1 and C2 are also complex. You can rearrange it as
f(x) = (C1 + C2) cos(x) + i (C1 - C2) sin(x)
or
f(x) = A1 cos(x) + A2 sin(x)
wherne A1 and A2 are complex constants.
If course if you want to restrict f(x) to be a real function, A1 and A2 must be real. That condition is the equivalent to C1 and C2 being complex conjugates, so that C1 + C2 is real and C1 - C2 is imaginary.