Numerical Methods (MATH 254) Assignment -2 Sunday, April 03, 2016 1. Solve the following initial value problem using the modified Euler method With h = 0.1 for x [0, 0.3]. y ' = y + x , y (0)=1 . Compare with the exact solution x y (x)=2 e - x - 1. ' 3 2. Given y =x + y , y (0)=2 , compute y (0.2) and y (0.4) using the Runge-Kutta method of fourth order. 3. Use the classical Runge - Kutta formula of the fourth order ti find the numerical solution at x=0.8 for dy = x + y , y ( 0.4 )=0.41 . dx Assume the length h=0.2 . 4. Solve the initial value problem for a damped mass- spring sytem y ( 0 )=3, y ' ( 0 )=2.5 y ' ' +2 y ' + 0.75=0, By the Euler method for the systems with h=0.2 for x=0 to 1 (where x is time) 5. Solve the initial value problem '' y = xy , y ( 0 )= 1 2 3 2 3 . 3 () =0.35502805 1 ' y ( 0 )= 1 3 ( ( )) 1 3 . 3 =0.25881940 By the Runge - Kutta method for the systems with h=0.2; do5 steps . 6. Find the solution of the boundary value problem '' y = y+ x , x [ 0,1 ] , y ( 0 )=0, y (1 )=0 With the shooting method. Use the Runge - Kutta method of the second order to solve the initial value problems with h=0.2. 7. Use the shooting method to solve the mixed value problem '' x u =u4 x e , 0< x <1, u ( 0 )u ' ( 0 )=1, u ( 1 )+u ' ( 1 )=e . Numerical Methods (MATH 254) Assignment -2 Sunday, April 03, 2016 8. Find difference approximations of the solution y(x) of the boundary value problem x 2 sin y ' ' +8 y ( 0 )= y ( 1 )=0 1 Taking step- length h= 4 and h=1/6 . Also find an appropriate value for ' y (0 )