Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-005 · ★☆☆ Basic

If \(f\) and \(\partial f/\partial y\) are continuous near \((x_0,y_0)\), the theorem ensures that the initial value problem:

Choose one option.

Show answer

B. has a unique solution on some interval around \(x_0\)

Explanation

The result is local: a unique solution exists on some (possibly small) interval around \(x_0\); nothing guarantees it extends to the whole line.

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Review the concept: Existence and uniqueness theorem

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.

\[f,\ \frac{\partial f}{\partial y}\ \text{continuous near }(x_0,y_0)\ \Longrightarrow\ \text{a unique solution}\]

More practice problems: existence and uniqueness theorem

ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-001 For y'=xy^2, what is ∂ f/∂ y?
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-002 For which initial value problem does the theorem guarantee a unique solution?
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-003 The problem y'=√y,\ y(0)=0 has the solution y=0. Which of these functions, for x≥ 0, is another sol…
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-004 For y'=(y)/(x-3), where do f and ∂ f/∂ y stop being continuous?
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-006 For y'=x+y^2, can two different solutions cross?
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-007 The solution of y'=y^2,\ y(0)=1 is y=(1)/(1-x). What is the largest interval containing 0 on which…
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-008 For y'=3y^2/3, the set of initial points (x_0,y_0) where the theorem guarantees a unique solution i…
ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-009 The hypotheses of the theorem fail for an initial value problem. What can be concluded?