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Unformatted text preview: Ma 2a P: Homework N.2
due Tuesday Oct 19, 12 noon
1. For the following diﬀerential equations describe the equilibrium solutions and the asymptotic behavior of the other solutions, for diﬀerent choices of the initial condition y (0) = y0 : dy = ey − 1, with initial conditions −∞ < y0 < ∞; dt dy • = y 2 (1 − y )2 , with initial conditions −∞ < y0 < ∞; dt dy • = y (a − y 2 ), for diﬀerent possible values of the parameter a > dt 0, a = 0, or a < 0, and with initial conditions −∞ < y0 < ∞. •
dy 2. Check if the following equations of the form M (x, y )+ N (x, y ) dx satisfy the condition My = Nx . If so, ﬁnd a function ψ (x, y ) such that ψx = M and ψy = N . dy ax + by =− ; dx bx + cy y dy x • +2 = 0; 2 + y 2 )3/2 2 )3/2 dx (x (x + y • • (yexy cos(2x) − 2exy sin(2x) + 2x) + (xexy cos(2x) − 3) dy = 0. dx 1 3. For each of the following equations, ﬁnd an integrating factor so that the resulting equations can be solved by the method of the previous problem, then write the solutions, as curves in the (x, y )plane. x dy • 1 + ( − sin(y )) = 0; y dx dy = 0. • y + (2xy − e−2y ) dx 4. Write the following linear diﬀerential equations equivalently as integral equations, and equivalently as ﬁxed point problems φ(t) = T (φ(t)); starting with ψ (t) ≡ 0, compute the iterates T k (ψ (t)) and show that they converge to the solution φ(t) = T (φ(t)). • y = −y − 1 with y (0) = 0; • y = y + 1 − t with y (0) = 0. 2 ...
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 Spring '10
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