Math 129 Section 005H Lecture 39: Taylor series solutions to differential equations (Ch. 11 supplement)¹¹ 1 This document is licensed under a Creative Commons Attribution 3.0 United States License Math 129 Section 005H Lecture 39: Taylor series solutions to differential equations (Ch. 11 supplement)¹¹ 1 This document is licensed under a Creative Commons Attribution 3.0 United States License Alternative method

Taylor series and differential equations

(Monday, November 22, 2021)

Today I covered variations of Examples 3(b) and 5 from the Ch. 11 Supplement. We end with this:

Theorem: Taylor series solutions of differential equations

If $p(x),q(x),r(x)$ have convergent Taylor series in an interval $-R<x<R$ for $R>0$ , then the solution to the initial value problems²² 2 The phrase “initial value problem” refers to a differential equation together with enough initial conditions to uniquely determine a solution, i.e., a first-order initial value problem means a first-order differential equation with one initial condition, a second-order initial value problem means a second-order differential equation with an initial value and an initial derivative, etc.

\frac{dy}{dx}+p(x)y=q(x)~{},~{}~{}y(0)=a

and

\frac{d^{2}y}{dx^{2}}+p(x)\frac{dy}{dx}+q(x)y=r(x)~{},~{}~{}y(0)=a~{},~{}~{}y^% {\prime}(0)=b

can be expressed as a Taylor series that also converges on $-R<x<R$ :

y(x)=C-0+C_{1}x+C_{2}x^{2}+C_{3}x^{3}+\cdots.