The following problem is from Application 6.3 from page 412, off the book \"Diff
ID: 2980978 • Letter: T
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The following problem is from Application 6.3 from page 412, off the book "Differential equations and Boundary Value Problems" 4th ed. C. Henry Edwards and David E. Penny I had p = 8, q = 9, a= 1, b= 2 Q: How do I take the following differential equations and convert them into there perspective trigonometric functions? dR/dt = (.08)-(.0001)RF and dF/dt = -(.09)F + (.0002)RF The answer I'm looking for is something along these lines: R(t) = a*C1cos(wt) + b*C2sin(wt) and I think F(t) = [R' -(.08)] / [-(.0001)R] a = ? b = ? C1 = ? C2 = ? First I got critical points (0,0) and (450,) From (450,800) I got Eigen values of + or - (0.084853) so 0.08ia - 0.045b = 0 and .16a + 0.08ib = 0 approximately yields a = 1 and b = 2. So what I did was: (1)C1cos(.08t) + (2)C2sin(.08t) = R(t) and R(0) = 450 so 450 = C1 but then I get F(t) = 800 - (1/R)(10000)(R') and F(0) = 800 and this yields C2 = 0, but C2 can not equal 0, that does not make any sense.Explanation / Answer
R(t) = a*C1cos(0.8t) + b*C2sin(0.8t)
=450cos(0.8t) + 2*C2sin(0.8t)
R(0) =450
R'(t) = -450wsin(0.8t) + 2C2*0.8*cos(0.8t)
R'(0) = 2C2*0.8
now
F(t) = [R' -(.08)] / [-(.0001)R]
F(0) = [R'(0) -0.8]/[-0.0001R(0)]
8000 = [2C2*0.8 -0.8]/[-0.0001*450]
8000*-0.0001*450 = [2C2*0.8 -0.8]
360 = (2C2-1)*0.8
C2 = 225.5
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