Math 231, Spring 2008- HW set 10 (due 4/24) (some problems
are from [Tenenbaum-Pollard])
1. The following first-order equations for y=y(t) are of ‘homogeneous
type’. Find the ‘general solution’, or find the solution to the initial-value
problem (in implicit form if necessary).
(i)ty0−y−tsin(y/t) = 0 Ans.y = 2tarctan(Ct)
(ii)ydt +tlog( y
t)dy −2tdy = 0 Ans.y =C(1 + log t/y)
(iii)y0=ty −y2
t2, y(1) = 1 Ans.y =t/(1 + log t), t > 0
(iv)(t2+y2)dt = 2tydy, y(−1) = 0 Ans.y =pt2+t, t ∈(−∞,−1]
2. For the following equations, state for which y0the given initial-value
problem is solvable, and what the interval of definition of the solution would
be. (It is possible the answer can’t be more precise than ‘some open interval
including t0’).
(i)ty0−y−tsin(y/t) = 0, y(1) = y0
(ii)(2t2+ 1)y00 −4ty0+ 4y= 0, y(0) = y0
(iii)t2y00 +ty0−y=t2e−ty(1) = y0
(iv)ydt +tlog( y
t)dt −2tdy = 0 y(1) = y0
3. For the following second-order homogeneous equations (y=y(t)),
a solution is given. Use the ‘reduction of order’ method to find a second,
linearly independent solution.
(i)2t2y00 + 3ty0−y= 0, y1=√t
(ii)(2t2+ 1)y00 −4ty0+ 4y= 0, y1=t
4. For equation (i) in problem 3, reduce to ‘standard form’ (where the
coefficient of y00 is 1), then apply the Liouville transformation:
y(t) = z(t)e−1
2P(t), P (t) = Zp(t)dt,
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