Suppose y1=-6e^3t-2t^3, and y2=8e^-9t-2t^3 are both solutions of a certain nonho

Question

Suppose y1=-6e^3t-2t^3, and y2=8e^-9t-2t^3 are both solutions of a certain nonhomogenous equation: y"+p(t)y'+q(t)y=g(t) (A) Is y=18e^3t -4e^-9t + 7t^2 also a solution of the equation? (prof told me I can solve a couple of ways...superposition principle+noting the structure of a homogenous vs non homogenous eq...which are "subtly different". He said that won't help with parts b or c though. (B) What is the general solution of the equation? (C) Determine p(t), q(t), and g(t) in the equation above. I have no idea how to do this...and neither do several people who are math tutors, so any help is appreciated. Please try to explain the concepts here, as this is mainly a conceptual question that I'm having trouble pulling together.

Explanation / Answer

A.

-0.5*y2-3*y1 = 18e^3t -4e^-9t + 7t^2 =>y=18e^3t -4e^-9t + 7t^2 is also a solution of the equation.

B. General solution = m*y1+n*y2 , where m and n are real numbers.

C.

STEP 1:

Get (y1)' and (y1)'' and put in thenonhomogenous equation: y"+p(t)y'+q(t)y=g(t). Then this will give you FIRST equation involvingp(t), q(t), and g(t).

STEP 2:

Get (y2)' and (y2)'' and put in thenonhomogenous equation: y"+p(t)y'+q(t)y=g(t). Then this will give you SECOND equation involvingp(t), q(t), and g(t).

TILL NOW YOU GOT 2 EQUATIONS AND THREE UNKNOWNS.

STEP 3:

Get (y)' and (y)'' and put in thenonhomogenous equation: y"+p(t)y'+q(t)y=g(t). Then this will give you THIRD equation involvingp(t), q(t), and g(t). HEREy=18e^3t -4e^-9t + 7t^2.

NOW YOU HAVE THREE EQUATIONS AND THREE UNKNOWNS. SOLVE THEM TO GETp(t), q(t), and g(t).

I HOPE THIS EXPLAINS THE CONCEPT.

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