Question

An equation of the form y'px)ygx)y", a 0,1 is called the Bernoulli equation If we divide by ya we get g(x) . у Чу' + yay' p(x)y-a Next let us make the substitution u y *) (i) Show that y ay' u' [3] = 1-a (ii) By substituting in (*) show that u' + (1— а)p(x)и %3D (1 — а)g(хx) [3] We now have a linear ODE that can be solve for u(x) using an integrating factor. We can then use (**) to obtain y(x) (iii Solve the following Bernoulli equations (a) x2у' + 2ху — уз %3D 0, х>0 [10] (b) y' ey ay3,e > 0,0 > 0 (This equation occurs in the study of the stability of fluid flow.) [10]

Answer 1

Given that a gltzay - y =0 @ Divide it by n we get s'ta ja ne go - compare it by y't play ý =3(4) oyo. :þcu = 3:30) - Rs. Let us g -d be the substitution, 3 3 ☺ U = tz = 5=u . = y0 - 129. des ys 1 from 6 5 s'tj 2 - , we get utaus wegt multiply with -2, u It is which of the is a form u't p (a) u = Q(a). linear differential equation

Sdk IF e - = mina of en ② is . The genera solation 4. (8-F) = false) dr. a C - ] = 3 2 2 2 and the =-2 fede astu - s + c M५

= 3 +c: Have u = sz yaxy = Suste multiply with a", wagt which is the general solution of giren ☺ consider y'=&y-ayo compare with y't pca) y = g(x).ya 223, bca --€, 900=-0. Lt uzsa, e eu sou ' = {y-oy' n jou beer → z du = ue -- - -2e utgo here al du tae uzzo. {fo are positive constants. It in of the form du + Plazua Q (4) I.F = {punde - Jazde - rex Thus the general solution is u. (T-F) = 1 an (4) dx

-sende wiesze eextc BEX, we we - 2EX Et ceex then u:52. Feste ēzex which is the generd solation of en