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By Jack K. Hale

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This time we must content ourselves with the solution expressed implicitly, since it is not feasible to solve for y in terms of x. • EXERCISES Determine which of the following equations, in Exercises that are exact. 1. ( �) x+ dy + y dx = 0 2 . ( sinx tany + l)dx + cosx sec2 y dy =0 1-19, is exact. 5 Exact Equations 21 3. (y-x3)dx+ (x+ y3)dy = 0 4. (2y2-4x+ 5) dx= (4- 2y+4xy)dy 5. (y+ y cosxy)dx+(x +x cos xy)dy=0 6. cos x cos2 ydx+2 sin x sin y cos ydy=0 7. (sin x sin y-xeY)dy= (eY+cos x cos y)dx 8.

ORTHOGONAL TRAJECTORIES AND ..... FAMILIES OF CURVES We have already noted that it is useful to think of the collection of solutions of a first­ order differential equation as a family of curves. 9. We solved the differential equation eY dx+ (xeY +2 y ) dy = 0 and found the solution set X • eY + y2 = C. 24) For each value of c, the equation describes a curve in the plane. Conversely, if we are given a family of curves in the plane then we can produce a differential equation from which the curves all come.

19, with Io = 0 and E0 =fa 0, show that the current in the circuit builds (L ln 2)/ R seconds. up to half its theoretical maximum in 2. 37) for the case in which the circuit has an initial current the emf impressed at time (a) (b) 3. 37) and show that (a) Ohm's law is satisfied whenever the current is at a maximum or minimum. (b) The emf is increasing when the current is at a minimum and decreasing when it is at a maximum. 36) and if Q= 0 when t= 0, then find the charge buildup Q = Q(t) on the capacitor in each of the following cases: (a) E is a constant Eo (b) E= Eoe-r (c) E= E0 cos wt 4.

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