3.4 Equations with Functions as Unknowns 197with general solutionu=C 1 y^2 +C 2 y−^2. Remember thatu=p^2 =(y′)^2 , from which
we obtain the first-order differential equation
y′=±√
C 1 y^2 +C 2 y−^2 =√
C 1 y^4 +C 2
y.
This we solve by separation of variables
dx=±ydy
√
C 1 y^4 +C 2,
which after integration gives
x=±∫
ydy
√
C 1 y^4 +C 2=±
1
2
∫
dz
√
C 1 z^2 +C 2.
This last integral is standard; it is equal to 2 √^1 C 1 ln|y+
√
y^2 +C 2 /C 1 |ifC 1 >0 andto 2 √^1 |C 1 |arcsin(|CC^12 |y)ifC 1 <0 andC 2 >0. We obtain two other families of solutions
given in implicit form by
x=±1
2
√
C 1
ln∣∣
∣
∣∣y+√
y^2 +C 2
C 1
∣∣
∣
∣∣+C 3 and x=±1
2
√
−C 1
arcsin|C 1 |y
C 2+C 3 ,
that is,
x=Aln|y+√
y^2 +B|+C and x=EarcsinFy+G. Here are more problems.560.Solve the differential equation
xy′′+ 2 y′+xy= 0.561.Find all twice-differentiable functions defined on the entire real axis that satisfy
f′(x)f′′(x)=0 for allx.
562.Find all continuous functionsf:R→Rthat satisfy
f(x)+∫x0(x−t)f(t)dt= 1 , for allx∈R.563.Solve the differential equation
(x− 1 )y′′+( 4 x− 5 )y′+( 4 x− 6 )y=xe−^2 x.