Concept guide
Initial-value problem: auditing four polynomial candidates
Initial-value problem: y''(x)=6x on ℝ with y(0)=1 and y'(0)=2. Audit the four integer-coefficient polynomial candidates y_A=x³+2x+1, y_B=x³+1, y_C=x²+2x+1 and y_D=3x⁵−15x⁴+21x³+2x+1 one by one: check the equation residual and the two initial values separately, decide which candidate solves this initial-value problem, and explain why checking only the initial data, only one initial value, or the residual only on a finite grid is not enough.