Substitution Methods and Exact Equations
Dy/dx = F(y/x)
v = y/x, y = vx, dy/dx = v + x(dv/dx)
x(dv/dx) = F(v) - v
dv/dx + (1-n) P(x)v = (1-n) Q(x)
Criterion for Exactness:
F(x,y) = ∫ M(x, y) dx + g(y)
Verify that the given differential equation is exact; then solve it.
31. (2x + 3y) dx + (3x + 2y) dy = 0
Answer: x2 + 3xy +y2 = C
35. (x3 + y/x) dx + (y2 + lnx) dy = 0
Answer: 3x4 +4y3 +12ylnx = C
38. (x + tan-1y) dx + ((x +y) / (1 + y2)) dy = 0
Answer: x2 + 2xtan-1y + ln(1 +y2) = C
Differential equatiosn are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.