Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-006 · ★★☆ Intermediate
For \(y'=x+y^2\), can two different solutions cross?
Choose one option.
Show answer
- C. No, because exactly one solution passes through each point
Explanation
Since \(f=x+y^2\) and \(f_y=2y\) are continuous on the whole plane, exactly one solution passes through each point; if two curves crossed, there would be two solutions through that point.
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-005 If f and ∂ f/∂ y are continuous near (x_0,y_0), the theorem ensures that the initial value problem:
- 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?
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