Advanced book on Mathematics Olympiad

Real Analysis 593

for some real constantC. In particular,fis bounded.
(R. Gelca)

556.The idea is to write the equation as

Bydx+Axdy+xmyn(Dydx+Cxdy)= 0 ,

then find an integrating factor that integrates simultaneously Bydx+Axdy and
xmyn(Dydx+Cxdy). An integrating factor ofBydx+Axdy will be of the form
x−^1 y−^1 φ 1 (xByA), while an integrating factor ofxmyn(Dydx+Cxdy)=Dxmyn+^1 dx+
Cxm+^1 yndywill be of the formx−m−^1 y−n−^1 φ 2 (xDyC), whereφ 1 andφ 2 are one-variable
functions. To have the same integrating factor for both expressions, we should have

xmynφ 1 (xByA)=φ 2 (xDyC).

It is natural to try power functions, sayφ 1 (t)=tpandφ 2 (t)=tq. The equality condition
gives rise to the system

Ap−Cq=−n,

Bp−Dq=−m,

which according to the hypothesis can be solved forpandq. We find that

p=

Bn−Am

AD−BC

,q=

Dn−Cm

AD−BC

.

Multiplying the equation byx−^1 y−^1 (xByA)p=x−^1 −my−^1 −n(xDyC)qand integrating,
we obtain

1

p+ 1

(xByA)p+^1 +

1

q+ 1

(xDyC)q+^1 =constant,

which gives the solution in implicit form.
(M. Ghermanescu, ̆ Ecua ̧tii Diferen ̧tiale(Differential Equations), Editura Didactic ̆a
̧si Pedagogica, Bucharest, 1963) ̆

557.The differential equation can be rewritten as

ey

′lny

=elnx.

Because the exponential function is injective, this is equivalent toy′lny=lnx. Inte-
grating, we obtain the algebraic equationylny−y=xlnx−x+C, for some constant
C. The initial condition yieldsC=0. We are left with finding all differentiable functions
ysuch that

ylny−y=xlnx−x.