UDS MTH:311 Differential Equations 1 ST TRIMESTER 2012/13 DR. Y. I. SEINI 2012

Differential Equations Recall: A general solution is a family of solutions defined on some interval I that contains all solutions of the DE that are defined on I. In this chapter we will discuss finding general solutions of LDE of higher order than 1. We must investigate LDE’s.

Differential Equations For LDE – We will distinguish between: -Initial-Value and -Boundary-Value Problems

Differential Equations A LDE n-th order Initial Value Problem: Solve: Subject to: y(x 0 ) = y 0, y’(x 0 ) = y 1, …, y (n-1) (x 0 ) = y n-1

Differential Equations Theorem 4.11 Existence of a Unique Solution Let and g(x) be continuous on an interval I and let a n (x) ≠ 0 for every x in this interval. If x 0 is any point in I, then the solution y(x) of the IVP exists on I and is unique.

Differential Equations Example 1: The given family of functions is the general solution of the DE on the indicated interval. Find a member of the family that is a solution of the IVP.

Differential Equations A LDE n-th order Boundary Value Problem: Solve: Subject to: the dependent variable y or its derivatives are specified at different points.

Differential Equations So a 2 nd order Boundary-Value Problem: Solve: Subject to: y(a) = y 0, y(b) = y 1 Or y’(a) = y 0, y(b) = y 1 Or y(a) = y 0, y’(b) = y 1 Or y’(a) = y 0, y’(b) = y 1

Differential Equations In the case of BVP, even if the conditions are satisfied as in Thm , we may still have many, one or no solutions.

Differential Equations The given two-parameter family is a solution of the indicated DE on the interval (-∞, ∞). Determine whether a member of the family can be found that satisfies the boundary conditions.

Differential Equations Def: A linear n-th order DE of the form is homogeneous, and is non-homogeneous with g(x) ≠ 0.

Differential Equations To solve a nonhomogeneous equation like in (2), we must solve the associated homogeneous equation (1). We will state for the remainder of the section that - The coefficient functions a i (x) for each i, and g(x) are continuous and - a n (x) ≠0 for every x in the interval.

Differential Equations Superposition Principle – Homogeneous Eq. Let y 1, y 2, …, y n be solutions of the homogeneous nth-order DE (1) on an interval I. Then the linear combination y = c 1 y 1 (x)+c 2 y 2 (x) +…+c n y n (x) where c i are arbitrary constants, is also a solution on the interval.

Differential Equations Corollary’s 1)A constant multiple y = c 1 y 1 (x) of a solution y 1 (x) of a homogeneous LDE is also a solution. 2)A homogeneous LDE always possesses the trivial solution y = 0.

Differential Equations Def: Linear Dependence/Independence A set of functions f 1 (x), f 2 (x), …, f n (x) is said to be linearly dependant on an interval I if there exist constants c 1, c 2, …, c n, not all zero, such that c 1 f 1 (x) + c 2 f 2 (x) +…+ c n f n (x)=0 For every x in the interval. If the set of fn. Is not linearly dependant then its linearly independent.

Differential Equations Linear dependence means for example for a set containing two functions that one is a constant multiple of the other. Note that f(x) = x and g(x) = sin(x) are linearly independent, since neither is simply a constant multiple of the other.

Differential Equations Theorem Criterion for Linearly Independent Solutions. Let y 1, y 2, …, y n be n solutions of the homogeneous liner nth-order DE (1) on an interval I. Then the set of solutions is linearly independent on I iff W(y 1, y 2, …, y n ) ≠ 0 for every x in the interval.

Differential Equations What is the W(f 1, f 2, …, f n )=? If f 1, f 2, …, f n are fn such that each has at least n-1 derivatives. The determinant W(f 1, f 2, …, f n )= The Wronskian of the functions.

Differential Equations Recall the determinant computation:

Differential Equations Example: Determine whether the given set of functions is linearly independent on the interval (-∞,∞). a)f(x) = x, g(x) = x 3, h(x) = 3x 3 + 6x b)f(x) = 7, g(x) = sinx, h(x) = cosx

Differential Equations Def: Any set y 1, y 2, …, y n of n linearly independent solutions of the homogeneous linear nth-order DE (1) on an interval I is said to be a fundamental set of solutions on the on the interval. Thm There exists a f.s. of solutions for the homogeneous linear nth-order DE (1) on an interval I.

Differential Equations Theorem Let y 1, y 2, …, y n be a fundamental set of solutions of (1) on an interval I. Then the general solution of (1) on the interval I is y = c 1 f 1 (x) + c 2 f 2 (x) +…+ c n f n (x), where c i are arbitrary constants.

Differential Equations Example: Verify the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.

Differential Equations Reference: Differential Equations With Boundary-Value Problems Zill & Cullen Seventh Edition