Question

List four ways to solve a first order nonlinear differential equation in the order you would try them. What test would you apply to the ODE to determine if the method you were considering would work?

Answer 1

A1 09 + which is diff. esh. In r ardy, hence its solution There are four methods to solve first-ordes lineas differential equation, ů Equation solvable for p. (i) Equation solvable for a cillll Equation solvable for y (W) Lagrange's equation. il let first orden nth degree diff. esh.is rdy. ing + Pnel 22 ) & Photo - Whene P ,,P2, Ph functions of nearly. pn + Pipny + + Phyp+Pnzo where p= du 14-2) (P-22) (b-an) = where Q,, Qarannan are furetions of nardy. þ-Q;0 9 P-Q250 -Q0 þ-anzo and their solutions are Let $19,9, C,) =0, Hi(x, y, CEO Now general solution f,(9,9,637 401,9, c) fz (2.9,C) --- foly,y,c) =0 Wheie Ca(2= - Cnac clil for a u= fzly, ) t we can written as eh. o Now diff. woro to y it possible eliminate p otherwise Letiþ a parameter. , (9,9,6,150, F = $(9,6,5) I Fz (19, P, c) =0

[li) for y y egh. O can be written as y = $,(%) diff. woroto a dn Þ= 8.1*1,**) and put dy ap we get 11 on which is diff.egh. in panda, where p is dependent rniable and a is independent vantable, its solution File, P, c) = 0 we can elliminate ? bet p as a parametes. (iv) Lagranges eghy yox|) + FIN it is called lagrange's egna diff. wirito X pa Hp) + xf 'n on tft) de + [+-f(p)] = [xf9) F'(p) J Dk xf(p) + f (!) b- 5 (P) FCP) db p-of(p) -f1b which is lineas esh, ina and p. we can solve easily and its solutia x= $(bis) elliminate Þ and he get Solution y = $p, c) F (N) + Fto) du = da ap da f'(b)__*- form 6 and BE

we can choose ary method for solving differential egn.it egh. solvable for p the first methad we would apply afterwise others.