Question

the answer should be as computer answer

Consider the initial value problem: y' = 842+ y(0)=5. (y+5) Using TWO(2) steps of the following explict third order Runge-Kutta scheme ki = hf(nyn), k2 = hf(n+ihgyn+şkı), k3 = hf(en+h,yn+şk2), Yn+1 = yn +4(k1+3k3), obtain an approximate solution to the initial value problem at x = 0.6 Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a single five decimal digit number, for example 17.18263. YOU DO NOT HAVE TO ROUND YOUR FINAL ESTIMATE y(0.6) - Skipped

Answer 1

The problem is

$\frac{\mathrm{d}y }{\mathrm{d} x}=\frac{8x+5}{(y+5)^2}$, y(0)=5.

To obtain the value of y(0.6) after two steps, we have to take h such that after two steps, the value of x will be 0.6, i.e., 0+2h=0.6

=> h=0.6/2=0.3.

Then we first take xo=0. Then, You y Co)=5. Now, f(x, y) = du har Then, k, = hf (alo, %) = 0.3. f(0,5) 50.3 ( 866215) - 0.015 K2 = h f (xo + b got 4) +0.3/ 810 +03)+5 )= 0.0173826130 (5+0.015+5) K3=hf (60724 , yo +253) 20.3 / 860+2x0,+5 +0.07883415158 y = yota (k, +363) = 5+23 (K, +843) 25.062875614 So, Y:- 4 (0.3)- 5.062875614, where X; -0.- Then, "* = (2., 9,) - 0.315,445 ) 0.0219234433 K2 = hf (26+ + 0.3182,+ 8 +5 -0.02425829899. (4,+ ķj +5)2 K3=hf (2, + 21, 4, +2K2) 0.3187,7 +S) 20.02657815032. Cyet 2k2 +5) 92= 4o7*(K. +3+3) = 5.088290088 So, y2 = y(0.6)=51088290088, where x2=0.6 1:4 (0.6) =5088290088 3