|Sep11-12, 10:37 AM||#1|
Question about solving ODE with Complex eigenvalue
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
|Sep11-12, 12:23 PM||#2|
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)
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.
|Sep11-12, 03:42 PM||#3|
AlephZero, thank you! Really helpful!
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