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

348 Differential Equations (Ch. 9
z
2945. 2y' + (y'
2946. y'y*+yif */'
2947. 2jr-3/ i = 40 I
Gx)-y" = Q\ y = 0, y' = 2 for A: = 2.
2 = 0; y=l, #' = 2 for * = 0.
; t/= 1, */' -0 for jc = 0.
2948. 2yy"-\-y* */'* = 0; #=1, y' = l for * = 0.
2
2949. I/"-*/' -*/; y=-l, y' = l for x=l.
2950. / + -
i e>y-2^' i = 0; //-I, y' = e for ^--
2951. H-//y"-f y' = 0; = 0, 0' = 1 for*=l.
2952. (H-00')/ = (l+0'")0'; 0=1, 0'=1 for x =
2953. (*+l)0" + x0 ff = 0'; 0=-2, 0'=4 forx=l.
Solve the equations:
2954. y' =
2955. 0' = j
2956. y" /2 = 4y //
.
2957. --=
yy'y" y'* + . y"* Choose the integral curve passing through
the point (0, 0) and tangent, at it, to the straight line y + x = Q.
2958. Find the curves of constant radius of curvature.
to
2959. Find a curve whose radius of
the cube of the normal.
curvature is proportional
2960.
normal.
Find a curve whose radius of curvature is equal to the
2961.
normal.
Find a curve whose radius of curvature is double the
2962. Find the curves whose projection
vature on the //-axis is a constant.
of the radius of cur-
2963. Find the equation
on the assumption
of the cable of a suspension bridge
that the load is distributed the projection of the cable
uniformly along
on a horizontal straight line. The
weight of the cable is neglected.
2964*. Find the position of
sile thread, the ends of which
equilibrium of a
are attached at
flexible nonten-
two points and
which has a constant load q (including the weight of the thread)
per unit length.
2965*. A heavy body with no initial velocity is sliding along
an inclined plane. Find the law of motion if the angle of inclination
is a, and the coefficient of friction is p,.
(Hint. The frictional force is ji/V, where ^V is the force of reaction of the
plane.)
2966*. We may consider that the air resistance in free fall
is proportional to the square of the velocity. Find the law of
motion if the initial velocity is zero..