Calculus: Analytic Geometry and Calculus, with Vectors

210

This gives the first and hence the second of the formulas

e kxy = f1, y = Aekx.

Integrals

More complete treatments of these matters are given in textbooks on differential

equations.

11 After having digested the preceding problem (which required that some

ideas and methods be absorbed), find the solutions of the following differential

equations that satisfy the given boundary conditions

(a) dz=y,y=1when x=0 fins.:y=ex

(b) dx=2y,y=3when x=4

(c) dx=y,y=Owhenx=0

12 Fill in the right members of the three formulas

d-x dx

ate - dt

when

x =?

,Ins.: y= 3e2cx-9)

Ans.: y = 0

(a) dt2 = g fins.: g, gt + ci, -ITgtl + cit + c2

(b) x = gt2 + cit + c2 Ans.: Same as (a)

(c) d x = sin t fins.: cos t, sin t, -cost + c

(d) dt = cos 2t 4ns.: -2 sin 2t, cos 2t, "W sin 2t + c

(e)dx= e2t e4ns.: 2e2t, e2d, ,-e2t + c

dt

13 A particle P moves in the xy plane in such a way that its acceleration a

(a vector) is always -gj. Thus

a=0i-gj.

Show that its velocity and displacement vectors must be

v=cii+(-gt+kl)j

r = (cit + cz)i + (-gt2 + kit + k2)j,

where the c's and k's are constant. Find the equation of the path in rectangular

coordinates when ci 0 0 and again when ci = 0. Hint: To solve the last part,

put r = xi + yj so that

x = Cit + c2, y =-.lgt2 + kit + k2

and eliminate t.

14 This problem requires us to think about indefinite integrals and gives our

first glimpse of the famous and important formula for integration by parts.