Consider the initial–value problem with on the interval
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Consider the initial–value problem dxdy=y−x2 with y(0)=1 on the interval 0≤x≤2. Two different numerical experiments are proposed to approximate the value of the solution at x=2.
• Experiment I partitions the interval into ten uniform subintervals of length 0.20.
• Experiment II uses the non-uniform partition {0,0.05,0.15,0.30,0.50,0.80,1.20,1.55,1.80,1.95,2.00}.
Answer the questions that follow, using clear reasoning and correct mathematical terminology.
(a) Describe qualitatively how the magnitude of dxdy on 0≤x≤2 influences the local truncation error generated on an individual subinterval of length h. In your explanation, relate the error to both the step length and the steepness of the solution curve.
(b) Explain why the choice of shorter subintervals near x=0 and x≈2 in Experiment II can be expected to reduce the cumulative (global) error compared with Experiment I, even though both experiments contain the same total number of nodes.
(c) Suppose the exact value of the solution at x=2 is later found to be 0.718. Without performing explicit numerical calculations, compare the sign and likely relative size of the approximation error produced by Experiment I versus Experiment II, and justify your answer.