MATH 2574H Lecture : Discussion 262014.pdf
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Homogeneous substitution y" = f(y/x) xaxby"=axcyd + bxeyf; where a + b = c + d = e + f. ---> divide by xa+b -------> ((xaxb)/(xa+b )) y"=a(xcyd/xa+b ) + b(xeyf/xa+b ) (y/x)by" = (yb/xb)y" = ayd/xd. Exercise 3 y + 2xe-yx - xy" = 0 y" = y/x + 2e-yx v = y/x ----------> dy/dx = v + x(dv/dx) v + x(dv/dt) = v + 2e-yx; 2e-vx^2 = x(dv/dx) dv/dx = 2e-vx^2/x if the exponent were y/x y" = y/x + 2e-(y/x); v + 2e-v = v + x(dv/dx) M(x,y) + n(x,y)(dy/dx) = 0 want integration factor : exists if. = eintegrate[r(x)dx] or eintegrate[q(y)]; above stuff is not in textbook or. Calciv page 1 (3x + 2y) - (2x + 3y) = 0. N = 3x + 2y; m = 2x + 3y (tanx)y" = y + 1 y"/(y + 1) = cotx.
