194 3 Real Analysis
553.A not uncommon mistake is to believe that the product rule for derivatives says that
(f g)′=f′g′.Iff(x)=ex
2
, determine whether there exists an open interval(a, b)
and a nonzero functiongdefined on(a, b)such that this wrong product rule is true
forfandgon(a, b).
554.Find the functionsf, g:R→Rwith continuous derivatives satisfying
f^2 +g^2 =f′^2 +g′^2 ,f+g=g′−f′,and such that the equationf =ghas two real solutions, the smaller of them
being zero.555.Letfandgbe differentiable functions on the real line satisfying the equation
(f^2 +g^2 )f′+(f g)g′= 0.Prove thatfis bounded.556.LetA, B, C, D, m, nbe real numbers withAD−BC =0. Solve the differential
equation
y(B+Cxmyn)dx+x(A+Dxmyn)dy= 0.557.Find all continuously differentiable functionsy:( 0 ,∞)→( 0 ,∞)that are solu-
tions to the initial value problem
yy
′
=x, y( 1 )= 1.558.Find all differentiable functionsf:( 0 ,∞)→( 0 ,∞)for which there is a positive
real numberasuch that
f′(a
x)
=
x
f(x),
for allx>0.559.Prove that if the functionf (x, y)is continuously differentiable on the wholexy-
plane and satisfies the equation
∂f
∂x
+f∂f
∂y= 0 ,
thenf (x, y)is constant.