Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 13
Uniqueness of Solutions*
Theorem 13.1. Uniqueness of Solutions. Suppose that y = φ(t) is a
solution to the initial value problem
}
y ′ (t) = f(t, y)
y ′ (13.1)
(0) = t 0
where f(t, y) and ∂f(t, y)/∂y are continuous on a box R defined by
t 0 − a ≤ t ≤ t 0 + a (13.2)
y 0 − b ≤ y ≤ y 0 + b (13.3)
The the solution y = φ(t) is unique, i.e., if there is any other solution
y = ψ(t) then φ(t) = ψ(t) for all t ∈ R.
The following example illustrates how a solution might not be unique.
Example 13.1. There is no unique solution to the initial value problem
y ′ (t) = √ }
y
(13.4)
y(1) = 1
Of course we can find a solution - the variables are easily separated,
∫
∫
y −1/2 dy = dt (13.5)
2y 1/2 = t + C (13.6)
y = 1 4 (t + C)2 (13.7)
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