Answer all the problems.
problem1) Use Euler method to solve numerically the initial value problem
dx/dt= -2t u^{2}; u(0) = 1
Also solve the above initial value problem by fourth order Runge-Kutta method.
problem2) (i) Find out the root between 2 and 3, of the equation
x sin x + cos x = 0
(ii) Solve following systems of simultaneous linear equations using Gauss elimination method
(i) x_{1}+x_{2}+x_{3}=6
x_{1}-x_{2}-x_{3}=6
2x_{1}+x_{2}+7x_{3}=12
(ii) 2x_{1}+4x_{2}+6x_{3}=16
2x_{1}+4x_{2}+9x_{3}=8
4x_{1}+3x_{2}+2x_{3}=23
problem3) Functions below have roots in the intervals that are specified on their right sides. find out the roots of these functions in each of the intervals to an assured three important figures by use of bisection method:
(i) e^{x}- 3x, on (0,1) and (1,2)
(ii) x^{2} - √x -2 on (1,2)
(iii) lnx+ √x- 2 on (1,2)
problem4) (i) Determine real root of the equation 4x^{3} + 3x^{2} - 4x - 5= 0 using bisection method. Perform three reiterations.
(ii) If π = 22/7 is approximated as 3.14, determine the absolute error, relative error and relative percentage error.
problem5) Find smallest positive root of e^{-x}-cos(x)=0 by fixed point method.