Question

alpha = 3

beta =2 Can you solve it in a hour please Thank you very much.

1. Consider the following initial-value problem. y' = e(1+B)t ln(1 + y2), 0<t<1 y (0) = a +1 a) ( 15p.) Determine the existence and uniqueness of the solution. b) (15p.) Use Euler's method with h = 0.25 to approximate the solution at t=0.5. {

Answer 1

Answer = the answer is given below

Ang: Given Initial Value problem Sy'= e(tBt In (1+y2); ost! yoo)= 2+1 where a=3 ; p=2 (a) We have to determine the existence & Uniqueness of the solution. ye!= f(ty)= e34 ln (1+y2); osts/ y (0)=4 f(ty) is Continuous in (0,4). &]f(+8)) = |e3* ln (1782) <leselllmct837 ; ostsi & (0,4) ; where is a positive integer. the df dy est 24 (1+y2) at (0,4) tu 2x4 df dy (0,4) (1416) - ol 17 Jy 18,4) idf SK ; Where k is positive integer. Hence given initial value problem have exists & unique Solution. (b) We have to ase Euler's Method with h= 0.25 then find y(0:50) Here too ti = 0.25 i ta=0.50 ; L2= Yo=4 4.71 : = yo th ft., 40) (to, Jo) = (0,4) Ji= 4+ (0-25) f(0,4) = 4+ (0-25) [e36 ln (1+16)] = 4+ (0-25) In (17) = 4+ (0-25) (2-83) Y (to, yu) (0:25, 4071) Y2 = Yith f (4.9) = 4.71 +(0.25) f(0.25,4.71) (4-71) + (0:25) In(1+22 (4.2)) + (0.25) [(3:12) (3:14)] Y₂ = 5.67 (+2142) = (0.50, 5067) Hence lylo.so) 0.75 22-1841)] = 5.67 on's Scanned with CamScanner CS Scanned with CamScanner