Problem 1
Yes; f=t+y and f_y=1 are continuous everywhere.
Faculty use
Existence, Uniqueness and Maximal Intervals
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
The derivative with respect to y is not bounded/continuous near y=0; multiple solutions can remain at zero and then depart.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using continuity and local Lipschitz conditions to judge whether an IVP has one local solution. Use the formulas y′=f(t,y); f and ∂f/∂y continuous near (t₀,y₀) ⇒ local uniqueness.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to motion, heat flow, circuits, fluids, population change, and engineering balance laws, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.