Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-007 · ★★☆ Intermediate
The solution of \(y'=y^2,\ y(0)=1\) is \(y=\dfrac{1}{1-x}\). What is the largest interval containing \(0\) on which it is defined?
Choose one option.
A \((-\infty,1)\) B \((-\infty,\infty)\) C \((-1,1)\) D \((0,1)\) Show answer ← Back to the question
Explanation The function is undefined at \(x=1\). The largest interval containing \(0\) that avoids it is \((-\infty,1)\): the existence given by the theorem is only local.
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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