Exercise · ZW-EXISTENCE-AND-UNIQUENESS-THEOREM-002 · ★☆☆ Basic
For which initial value problem does the theorem guarantee a unique solution?
Choose one option.
Show answer
- C. \(y'=x+y^2,\ y(0)=0\)
Explanation
Only for \(y'=x+y^2\) are \(f=x+y^2\) and \(\partial f/\partial y=2y\) continuous near \((0,0)\). In the others the partial derivative with respect to \(y\) is unbounded at \(y=0\).
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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