Question

The differential equation (D4
+ 2D2
+ 1) y = x
2
cos x.

Answer 1

Answer:

The characteristic polynomial of the homogeneously equation

(D4+2D2+1)y=0

is given by p(t)=t4+2t2+1=(t2+1)2. p has the zeroes i and −i, both of order 2. Hence the general solution of (D4+2D2+1)y=0 is given by

c1cosx+c2xcosx+c2sinx+c4xsinx,

where c1,...,c4∈R.

A special solution ys of

(∗)(D4+2D2+1)y=x2

can be found by the "Ansatz" ys(x)=ax2+bx+c. Use (∗) to derive ys(x)=x2−2.

Hence the general solution of (D4+2D2+1)y=x2 is given by

c1cosx+c2xcosx+c2sinx+c4xsinx+x2−2,

where c1,...,c4∈R.

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Answer 2

Answer:

May b Now you are happy