SUMMARY
The differential equation y'=\sqrt{y} has multiple solutions when the initial condition is y(0) = 0. The solutions include y(t) = \frac{t^2}{4} and y(t) = 0 for all t. Picard's uniqueness theorem fails in this case because the function \sqrt{y} is not Lipschitz continuous at y=0, allowing for the existence of multiple solutions. However, if the initial condition y(0) > 0, the solution becomes unique as long as y remains positive.
PREREQUISITES
- Understanding of ordinary differential equations (ODEs)
- Familiarity with Picard's uniqueness theorem
- Knowledge of Lipschitz continuity
- Basic calculus, particularly differentiation and integration
NEXT STEPS
- Study the implications of Lipschitz continuity in differential equations
- Explore examples of non-unique solutions in ODEs
- Learn about the conditions under which Picard's theorem applies
- Investigate other forms of differential equations with discontinuities
USEFUL FOR
Students and educators in mathematics, particularly those studying differential equations, as well as researchers interested in the properties of solutions to ODEs.