Friday, 14 July 2017

complex analysis - Solving y+2y+2y=0: How to eliminate imaginary unit from solution?




y+2y+2y=0




(write characteristic equation)




λ2+2λ+2=0



(solve characteristic equation)



λ=1±i



(write general solution)



y=Ae(1i)t+Be(1+i)t




(apply Euler's formula)



y=Aet(cos(t)+isin(t))+Bet(cos(t)+isin(t))



(perform minor algebra/trig rearrangement)



y=(A+B)etcos(t)+(BA)etsin(t)i




Where do I go from here to eliminate the i? Plugging in the exponential formulas for sine and cosine leads back to the original general solution, with i's remaining.




Answer



Both (A+B) and (BA)i are arbitrary constants. Rename them C1 and C2 to get the general solution
y=C1etcos(t)+C2etsin(t)


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