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Home Differential Equations Existence and Approximation of Solutions Examples Example 4  Nonuniqueness  
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Example 4  Nonuniqueness
y' = 3y^{2/3}, y(0) = 0 has infinitely many solutions. The graphs split apart, as shown in Figure 14.4.6. Figure 14.4.6 Example 4 One solution is the constant solution y(t) = 0. A second solution is found by separating variables: ⅓y^{2/3} dy = dt, y^{1/3} = t + C, C = 0 y = t^{3}. This solution can be checked by differentiation: y' = 3t^{2} = 3(t^{3})^{2/3} = 3y^{2/3}. The other solutions go along the line y(t) = 0 in some interval and branch off the line y(t) = 0 to the right and left of the interval. The full list of solutions is: Here a is either a real number or ∞, b is either a real number or +∞, and a ≤ 0 ≤ b. In the case that a = ∞ and b = +∞, the solution is the constant function y(t) = 0. These solutions all have the same initial value y(0) = 0. The Uniqueness Theorem does not apply in this example because the function f(t, y) = 3y^{2/3 }has no derivative at y = 0, so that f(t, y) is not smooth at y = 0.


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