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Major Assignment

Question 1:

(i) 

so which is the only points at equilibra

(ii)

(linear system)

(iii)

Eigen vectors

Eigen set to  composition system :

solve to x’(t) =Ax(t)

so the solution are unique by picard, so

(iv)

At () :

(v)

 Eigen values :            

Eigen vector:

Real and distinct eigen vector

picard => x(t) =

  •  =

     =>

()+

therefore we have

we can also written this term of as    cos (h)

(vi)

Trajectory line : if y is small and x large and negative, solution diverge beyond the of both equilibrium. This teaches us an important lesson, that is the behavior of non linear system can be unknown even with complete knowledge of equilibrium region around these equilibrium.

Question 2:

There  

  • From h=1
  • From h=1/2
  •  

               Improvement


Question 3:

Laplace of this equation 

Now if

      same as  C=-1/2 and B=-1/4

X(s)=

Question 4:

(i)

f  is a rational function of polynomial where the element does not write it.

 (ii)

The differential equation is by solving this I have to find that this equation so this linear

(iii)

for X(0)=1

 (iv)

The picard

  • [f(x)]=
  • [f(x)}=[x]/[7+42t]   f/7 for

Therefore by picard a unique solution exists :

x:[0,T]  , where T 1/L = 1/1/7 =7

As this is uniform , at T=7k allow is to till [0,0], generating a solution T>0

(v)

x(n+1) =xn+h(x(n))/(1+42h)

  • x(n+1) =

note this input [N(n+1) < (1+h/7)^n+1

(vi)

x(n+1) =xn+h(x(n))/(1+42h)

(vii)

In which taking the invert of the part (v)

  • one step error ;{E} <

(viii)

The error below un h-> like

(1+h/7)^(2h+2)

for n, h lrge , the error below up to no matter have suffer h is i.e FE is never move for this DE.

this because the solution to this DE is   which FE have a lot of table opportunity

Question 5 :

f(x)= (1-x)(x-2)(x-3)^3

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