Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-009 · ★★★ Advanced
The hypotheses of the theorem fail for an initial value problem . What can be concluded?
Choose one option.
A That no solution exists B The theorem concludes nothing: there may be one solution, none, or several C That there is more than one solution D That a solution exists but is not unique Show answer B. The theorem concludes nothing: there may be one solution, none, or several← Back to the question
Explanation The theorem gives sufficient conditions, not necessary ones. If they fail, the problem must be studied by other means: for example \(y'=\sqrt{y}\), \(y(0)=0\) has several solutions, but other problems with failed hypotheses have only one.
The existence and uniqueness theorem guarantees that an initial value problem has a unique solution near the initial point, if the equation and its partial derivative with respect to the unknown are continuous there.
\[f,\ \frac{\partial f}{\partial y}\ \text{continuous near }(x_0,y_0)\ \Longrightarrow\ \text{a unique solution}\]
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