Suppose b24mk0 Prove all solutions yt of the ODE mybyky0 wil

Suppose b²-4mk=0. Prove all solutions y(t) of the ODE my\"+by\'+ky=0 will converge to zero as t increases, that is lim_t-->inf y(t) = 0.

Solution

It is a linear homogeneous equation with constant coefficients so solution is of the form

y=exp(rt)

Substituting gives

mr^2+br+k=0

General solutoin is

r=\\frac{-b+\\sqrt{b^2-4mk}{2},r=\\frac{-b-\\sqrt{b^2-4mk}{2}

Using given constraint b^2-4mk=0

r=-b/2

So repeated roots

So general solution is

y(t)=e^{-bt/2}(A+Bt)

If b>0 then e^{-bt/2}(A+Bt) goes to 0 as t tends to infinity and heence , y tends to 0 as t tends to infinity for all A and B