By Carl M. Bender

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31 Solve the following differential equations: = y/x + I/y; (b) y' = xy/(x' + y'); (e) y' = x' + 2xy + y2; (a) y' • 34 FUNDAMENTALS (0) (I) (E) Id) yy" 2IY')'; Ie) y' \I +x)Y'lx'; (1) X'y' + xy + y' 0; Ig) xl yll In x + In y); Ih) IX + i) + 2(i + y + x 1 )y' = 0, using an integrating factor of the form I(x, y) = e" +b,; (i) -xy'+y=xy'[Y(l) 1]; (j) y" [1 + xr'(y')' 0 [y(0) = l(O) = I]; (k) 2xyl + y' x' = 0; (I) y" (y')'e-' (if y' 1 at y = co, find y' at y = 0); (m) / Iy x I lif y(O) = t. find y(I)]; (n) xi y + xe'/'; (0) y' (x' 3x'y' y')/(2x'y + 3/x); (p) (x' + I)y' = xy, y(e) = e; (q) y" + 2y'y 0 [y(0) = l(O) = -I]; (r) x'y" + xy' = 3x' [y(l) = y(2) = 1]; Is) yl(y')'y" [y(O) = y'(O) = 1); (I) xy' Y + (u) (xy)y' + y y = 2xy [try an integrating factor of the form 1= I(y)]; (v) (x sin y + e")y' = cos y; (w) (x + y'x)y' + x'y' = 0 [y(l) = 1]; (x)(x I)(x 2)y'+y=2[y(0) I]; (y) y' I/(x + e"); (z) xl + y = y'x+.

Finally. this equation may be reduced to first order by letting w(v) v'(t): 3D' + w' + 6vw + vw dw du vw + 1 = 0. Again the final equation is very complicated. but it is the first-order equivalent of the third-order equation y" + yj 0, The reduction through two orders has followed the sequence of transformations autonomous (order 3) .... scale invariant (order 2)-equidimensional (order 2) .... autonomous (order 2) - complicated mess (order I). (c) Not all equations become too complicated to solve when their order is reduced, Thc equation in part (c) of the previous example simplifies beautifully when the transformation suggested by scale invariance is made.

We also recommend a more sophisticated variation by M. S. Klamkin called Ihe Great Snowplow Chase: "One day it started snowing at a heavy and steady rale. M. from the same place and all collided at the same time. " (Answer. ) Clue: The speed of the plow is inversely proportional to the height of the snow. 23 A man stands atop a mountain whose altitude is given by: = e-·x'Hy" and pours boiling oil upon the climbers below him. What paths do the rivulets of oil follow? 24 Show that a necessary and sufficient condition that M(x, y) + N(x, y) dy/dx = 0 be exact is oM/oy = oN/ox.