1. As we\'ve seen, to solve ODE of the form y\'\' + py\' + qy = 0; p and q const
ID: 3345205 • Letter: 1
Question
1. As we've seen, to solve ODE of the form
y'' + py' + qy = 0; p and q constants
we assume that the solution has the form e^rt, and then we plug this back into the ODE
to get the characteristic equation: r^2 + pr + q = 0. Given that this equation has a double
root r = r_1 (i.e., the roots are r_1 = r_2), show by direct substitution that y = te^rt is a
solution of the ODE, and then write down the general solution.
2. Suppose that z(t) = x(t) + iy(t) is a solution of
z'' + pz' + qz = Ae^iwt
Substitute z(t) into the above equation. Then compare (equate) the real and imaginary
parts of each side to prove two facts:
x''+ px' + qx = A cos wt
y'' + py' + qy = A sin wt :
Write a sentence or two summarizing the signicance of this result.
Explanation / Answer
the above explanation tells us that if we have complex solution of the form z(t) = x(t) + y(t)i, satisfying a differential equation whose constant is a complex number then the function x(t) will satisfy the same differential equation the only difference would be that the complex constant would be replaced by the real part of the same complex number. For example : e^iwt is replaced by coswt (which is real part of e^iwt). The function y(t) will also satisfy the same differential equation only the complex consant will be replaced by the imaginary part of the same complex number. For example : e^iwt is replaced by sinwt (which is complex part of e^iwt). [ Eulers law has been used to find out real and imaginary part of e^iwt
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