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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 $\pm i$ so general solution is $f(x) = c_1 e^{ix}+ c_2 e^{-ix}$ according to Euler's formula $$f(x) = c_1(\cos x+i\sin x) + c_2(\cos x-i\sin x )$$ it is different from the general solution generated by MAPLE why?

Thanks!

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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.

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