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

_Answers_467
x. 4- C* H
Xcosh -
|-C 2t where H is a constant horizontal tension, and = a. Hint*
= l^' ^\T)'
^2 = 77 y l
The differential equation 2965 2
' E( l uation of
=g(sina jiicosa). Law of motion, s^^- (sin a u, cos a) 2966. s ~ x
Xlncoshf/ I/ g ). Hint. Equation of motion, m-r^ mg \ r tn/ at*
k .
( -}
\"*/
2967. In
6.45 seconds. Hint. Equation
300 d~x
of motion, ~A~Z~ ^ y< 2968. a) No, b) yes>
c) yes, d) yes, e) no, f) no, g) no, h) yes 2969. a) t/'-{-y Q\ b) y" 2y' -f f/-=0;
c) A- 2/ 2x//'4-2f/-0, d) y'" 3y"+4i/~2//-0 2970. j/ = 3jt 5x f + 2x 8 . 2971. //-=
= (C,sm x-}-C2 cosx). Hint. Use the substitution // = */,. 2972. f/ = C,x-f
-fC 2 lnx. 2973. y = A + Bx 2
-{-x\ 2974. = --!- 4 * + ^L . Hint. Particular so-
O AT
the homogeneous equation r/j.v, / 2 --- . By
the method of the
v x 3
variation of parameters we find: C l ^~-\-A, C z -\-B 2975. y = A-\-
-| B sin x-\ Ccos x-Hn|sec x-f-tan ^I-f-sinx In cos | v| xcosx. 2976. = 2X
f/ C,e -h
H-C'jC 8* 2977. //^-C,e" 3* {-C 2e3V . 2978. // -=C t -|- C 2e x 2979. = */ C, cos x + C 2 sin x.
2980. //--e' v (CicosA'-l-C 2 Sin A-) 2981. y=~ e~** (C t cos 3v-|-C 2 sin 3x) 2982. y =
- r
(C^i-Cx)^-* 2983. i/-.--e- (C,e ]
~
2
+ C^~* rl ). 2984. If ft > 0, f/-
- CV*
2985. (/=*
* -r C,e" A
; if k < 0, j/-=C, cos J^^fex -[- C 2 sin
- * ---x / i/"TT
2
2
(C e t
-|-C 2 e
2
) 2986. //=.g 6 I C, cos J- x + C, sin
.//-4tf x + e lv . 2988. f/-=e~*. 2989. r/ = sin 2x. 2990. i/=l. 2991. // = a cosh .
2992. /y-0 2993. //=- C sin nx 2994. a) xe** (Ax* + Bx + C); b) 4 cos 2* h
-|-/Jsin2x; c) ^4cos2v + /J sin 2v + CvV A;
; d) g* (^ cos x + fl sin x), e) e x ;<
y(A\ s -\-B\i-C} 4- xe av (/; + ); f) xc* [(Av 2 -j- fiv + C) cos 2x+ (>x*-f x \-F)x
X sin 2xJ 2995. [/^(C.-j C a x) e ax '
+ 1 (2x + 4x -f- 3). 2996. y =^ (C ic
^-i-3A 2 . 2997. / = (C, -f C ax) P~ X -[- **.
7
2998. --. 'v
// C,c* I- C t>
2 + 2 2999. y - C,