If i have two conjugate roots as solution to 4th order ode, such that the two roots are proportional, how to find the four independent solutions of the equation?
I'm not sure what you mean by "two conjugate roots as solution to 4th order ode". Perhaps you mean two conjugate roots of the characteristic equation of the ode? But then I don't know what you mean by "the two roots are proportional". The characteristic equation of a 4th order ode is a fourth degree polynomial equation which generally has four roots. Just knowing two of the roots is not enough to tell you all four independent solutions. Or do you mean that two solutions are conjugates and there exist a second pair that are proportional to the first?
So two solutions are of the form a+ bi and a- bi and the other two are ak+ bki and ak- bki for some constant k?
If two solutions to the characteristic equation are a+ bi and a- bi then two independent solutions are and . The other solutions to the characteristic, ak+ bki and ak- bki, give and .
(The fact that the second pair of solutions to the characteristic equation are proportional to the first pair is not really of importance. If one pair of solutions is , and the other is l and then the four independent solutions to the differential equation are , , , and .