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Unformatted text preview: MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 28 Linear vs. Nonlinear Solutions
Example 1. Consider the initial value problem
t y + 2 y = 4 t2 , y (1) = 2. What is the largest region R such that there exists a unique solution? We solve this problem in three steps:
ﬁrst we ﬁnd the general solution to the diﬀerential equation, then we ﬁnd the particular solution using the
initial conditions, and ﬁnally we compute the region R by considering properties of this solution.
We begin by ﬁnding the general solution to the diﬀerential equation. (Recall that we solved this equation
in Lecture 5.) Upon dividing this equation by t, we see that
+ p(t) y = g (t)
The integrating factor is
µ(t) = exp where
t p(t) =
p(τ ) dτ = exp t g (t) = 4 t.
d τ = t2 .
τ Now multiply both sides of the diﬀerential equation by this function:
+ µ(t) p(t) y = µ(t) g (t)
+ 2 t y = 4 t3
t y = 4 t3
dt t2 · y (t) = t4 + C for some constant C .
Now we can ﬁnd the particular solution to the initial value problem. If we set t = 1 we ﬁnd the equation
2 = 1 + C . This gives C = 1, so that the solution to the initial value problem must be
y (t) = t2 + 1
t2 Note that this solution is deﬁned when t = 0.
Finally we compute the region R for which there does exist a unique solution to the initial value problem.
We can express the diﬀerential equation y + p(t) y = g (t) in the form y = G(t, y ), where
G(t, y ) = g (t) − p(t) y = 4 t − 2
t =⇒ ∂G
(t, y ) = − .
t These functions are both continuous when t = 0. The real plane R2 is divided in half by the line t = 0, so
our region R must be contained in one of these two. Also, the point (t0 , y0 ) = (1, 2) must be contained in
R, so the only possibility is the right-half plane:
R = (t, y ) ∈ R 0 < t < ∞, −∞ < y < ∞ .
Example 2. Now consider the initial value problem dy
3 t2 + 4 t + 2
2 (y − 1) y (0) = −1. Again, we compute the largest region R such that there exists a unique solution.
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.
- Spring '09