By Scarborough J.B.
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Let the given family of curves be described by the relation (7) F(x,y,c)=O. Differentiating this equation yields Fx + FyY' = 0, or y' = - Fx F . (8) y Next, we solve for c = c(x,y) from (7) and replace every c in (8) by this value c(x,y). Finally, since the slopes of curves which intersect orthogonally are negative reciprocals of each other, we see that the orthogonal trajectories of (7) are the solution curves of the equation , y = Fy F' x (9) Example 2. Find the orthogonal trajectories of the family of parabolas x=cy2.
12. y=cx 2 13. y=csinx IS. 8 Exact equations, and why we cannot solve very many differential equations When we began our study of differential equations, the only equation we could solve was dyjdt=g(t). We then enlarged our inventory to include all linear and separable equations. More generally, we can solve all differential equations which are, or can be put, in the form d dt>(t,y)=O (1) for some function >(I,y). To wit, we can integrate both sides of (1) to obtain that (2) >( t ,y) = constant and then solve for y as a function of 1 from (2).
In 1845 Verhulst prophesied a maximum population for Belgium of 6,600,000, and a maximum population for France of 40,000,000. Now, the population of Belgium in 1930 was already 8,092,000. This large discrepancy would seem to indicate that the logistic law of population growth is very inaccurate, at least as far as the population of Belgium is concerned. However, this discrepancy can be explained by the astonishing rise of industry in Belgium, and by the acquisition of the Congo which secured for the country sufficient additional wealth to support the extra population.
Differential equations and applications by Scarborough J.B.