Question

Problem 4. Find a function v(x) so that the substitution y(x) = u(x)+(x) transforms the differential equation y" + P(x)y' + Q(x)y = 0 into an equation for u of the form u" + f(x)u = 0. Write the function f in terms of P and Q. The last equation is called the normal form of a homogeneous linear second order equation.

Answer 1

Now + prasy' + gwy=0 substitute y(x)= uca Vla) - yeah - UCH) SV (X) + Vhen) Vad) y(x) = 4xV"ca) + u'(x) Vca) tu'ravl(a) + ul(x) V (n) 40 VW + Zuw v'ca) tuca Val Now UaV"()+2U' V'(a) + u'a vra + Pra) Cubo V'(a) + u'(a) v(a))+ Q(x) U(a) Ven) Now dividing by V(x) i we get U(a) + 4xV'() + 2 uc) vica) - V(a) V(x] + P(x) Ux) Vin) + P(a) u (a) t ara u(x)=0. = 0 vca) ullat l call, Provica) xu(a) Irla) v(a) + 2 ur var 1 + P(x) U'(x)=0 Vla)

Now we want above equation equal to. ut fra) url=0 this is possible only only if 24 2uch vm) + Praju'(a) = 0 va) tl 2 V (1) +P(x) = 0 Voa the greating dog vrar) = S- prx) dx - e prodot if then. we are taking real = è an then. 4(x) - uGura substitution transform y + proy't qruly=0 into u(x)t fra) un to where I fra - V"ca), Pra) V'(x) +ow von vow