Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

350_Differential Equations_[C/i. 9
we get
Solution. Solving the homogeneous equation
Hence, it may be taken that
*ir+if'=o f
y l = \nx and # 2 = 1
and the solution of equation (4) may be sought
in the form
Forming the system (3) and taking into account that the reduced form of
the equation (4) is t/"+~ = jt, we obtain
Whence
and, consequently,
+ >l and
y =
where A and B are arbitrary constants.
2968. Test the following systems
tionships:
a) x, x + 1; e) *, x\ x'\
2
b) x , 2x 2
; f) e* 9 e 2
*, e**\
(5)
of functions for linear rela-
c) 0, 1, x\ g) sin *, cos A:, 1;
2
sin x, cos 2
*, 1.
d) x, x+1, x +2; h)
2969. Form a linear homogeneous differential equation, know-
ing its fundamental system of equations:
sin x, y2 = cos x\
a) y l =
b) y^e*. y 2 ==xe*\
c) y^x* #2 = * 2
<
d) = ^
f/, x
y* = ^ x sin ^, = f/ ^ cos A:.
8
2970. Knowing the fundamental system of solutions of a linear
homogeneous differential equation
find its particular solution y that satisfies the initial conditions