Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-008 · ★★☆ Intermediate

For \(y'=3y^{2/3}\), the set of initial points \((x_0,y_0)\) where the theorem guarantees a unique solution is exactly:

Choose one option.

Show answer
  • D. the points with \(y_0\neq 0\)

Explanation

Here \(f_y=2y^{-1/3}\), continuous for \(y\neq 0\) and unbounded at \(y=0\). So the theorem applies only off the \(x\)-axis; on it (\(y_0=0\)) there are several solutions, such as \(y=0\) and \(y=x^3\).

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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

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