MATH 4B Lecture Notes - Lecture 5: Integral Curve, Piecewise Linear Function

• School: UCSB
• Department: Mathematics
• Course: MATH 4B
• Professor: Wei

Document Summary

Existence of solutions to the initial value problem - linear case. Existence and uniqueness guarantees that there is one and only one solution curve passing through an. Not every initial value problem has a solution. Some that do have solutions have more than one. Determine an interval in which the solution of the initial value problem is certain to exist: Existence and uniqueness theorem for linear ode = the solution of the initial value problem for linear equation exists, it is unique, and it is defined as long as the coefficients in the equation are defined. Given first order initial value problem (ivp) ode : Assume that and are given continuous functions defined on some interval. The solution is defined on the same interval. Given any initial time belonging to the interval and any value , we can find the constant in the general solution so as to satisfy the initial condition initial point: cannot exist when.