Set up the appropriate form of a solution yp of y^(3)+y'' = 4x^2 , but DO NOT determine the values of the coefficients.
guy this is what i have done so far
DE : y^(3)+y'' = 4x^2
CE: r^3 + r'' = 0
r1 = 0, r2= 0 , r3= -1
yc = C1 + C2x + C3 e^-x
now here i am confuse about what to do to guess yp. should yp be Ax^2?? plz help me understand this
If the rhs is some power of x then you would need to seek a particluar solution of the form
depending on the power . Now if any of your complimentary solution contains any of these terms then you need to raise the power by the number of terms that the complimentary solution contains. In your case it contains and so you need to "bump" up the particular solutions by 2.
This is third order linear non-homogeneous ODE which you have to seperate answers which is the and which the final solution would be the , and is the solution to the left hand side of the ODE and is the solution to the right hand side of the ODE. can be solved using the Laplace transformation.
i have not learn Laplace transformation yet. but i do know in order to solve this problem complete i have to use y(x) = Yc +Yp, Yc is easy to find since Yc is same in homogenous. but guess Yp is new thing for me. in every problem is different rule. what is the basic thing to follow here. what is it that i need to do to figure out Yp correctly?
oops.. i did it again, sorry for the wrong info
should be like this
,
is the solution to the left hand side of the ODE which is also same as your
and is the solution to the right hand side of the ODE that makes your ODE non-homogeneous.
to solve , you need to learn the (method of undetermined coefficients) or variation of parameters (if you have the basis of the solutions) which you need to find the Wronskian. But for this i suggest you to learn (method of undetermined coefficients) first.
thus,
substitute this and it's derivatives into the ODE, gives
rearranging the equation yields
by comparing coefficients for the both sides, for therefore
then
can anybody verify this.. i think it's wrong
maybe if we choose
it will be better?