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Question

Let y=y(t) be a solution of the differential equation y+2ty=t2, then 16limtyt is

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Solution

dydt+2ty=t2
I.F. =et2
Thus, solution is y.et2=t2et2 dt=12t.(2t.et2)dt
y.et2=t.et2212et2 dt+C
y=t2et2et22dt+Cet2
limtyt=12limtet22tet2+ct.et2=12

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