Introduction to Numerical Methods for Time Dependent Differential Equations

APPENDIX B

SOLUTIONS TO EXERCISES

SOLUTIONS FOR CHAPTER 1

1.3 Duhamel’s principle and integration by parts give the answer. One gets ₀ = y₀ in the resonance case and ₀ = y₀ − (−1)ⁿP_n⁽ⁿ⁾(0)/(μ − λ)ⁿ⁺¹ in the nonresonance case.

1.4

1.5 (a)

(b)

(c)

1.7 Assume that y(t) is a smooth solution that blowsup when t → T₀ < ∞. By the mean value theorem, for each t such that 0 < t < T₀, there exists a time , with 0 ≤ ≤ t, such that

Taking the limit t → T₀⁻, the right-hand side diverges and then dy/dt diverges at a time . But cannot be strictly smaller than T₀ because the differential equation forbids ...