Question

7. Consider these two boundary-value problems: r"-f (t, x, x') 1. Show that if x is a solution of boundary-value problem ii, then the function yt) x((t -a)/h) solves boundary-value problem i, where h b-a

Answer 1

gt)-a/h] , where h -b-a So, the LHS of the ean (i) becomes on substitution and, RHS beco mes h2 Also (using values From boundary conditions of egn tii) So, ylt) satispies the boundary conditions of eqn ti. Let, t-a/h u, say, then by 'avuh, in the eqn (1) Replace Also, replace 't by 'atuh' in the egn (2)

hỉ From eqn. (ii) h2 ‘t, 'u' ‘t' 's a dummy variable, so by Replace, as From eqns. (3) and (4), h2 Substitute in eqn. (5), above, to obtain y" = f ( t, y, y') shown af ter ean (2) y (a) ,y(b)-B boundary value solves the [It-a)/h] y(t) bus, problem given by e.i .e x": f ( t , x, x' )