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# Class Note for MATH 250A with Professor Lega at UA

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

Calculus and Differential Equations ll MATH 250 B Linear differential equations introduction amp overview a A linear ordinary differential equation of order n is an equation of the form dny dnily dy i dxn l an1xW 31x l 30Xy 7 hX7 where the coefficients a are functions of X o If the functions a are constant then the equation is said to have constant coefficients o If hx 0 then the differential equation is said to be homogeneous o A solution to the above differential equation is an n times differentiable function yx which satisfies the differential equation Definitions continued a An initial condition is the prescription of the values ofy and of its n 71st derivatives at a point X0 d dnil YX0 Yo7 TiXO Y177WIXo Ynilv where yo y1 yn1 are given numbers 0 Boundary conditions prescribe the values of linear combinations of y and its derivatives at two different values of x a We will see methods to solve linear differential equations Initial or boundary conditions should be imposed after the general solution of a differential equation has been found 0 Example 1 Consider y 7 2y y 0 o What is the order of this differential equation 0 Are y1x ex and y2X XEX solutions of this differential equation 0 Are y1x and Y2X linearly independent 0 Example 2 Consider 7xym l 2y cosx o Is the differential equation linear a What is its order o Is it homogeneous Does it have constant coefficients Existence and uniqueness of solutions 9 Theorem If the functions aX and hx are continuous on the interval 37 b and if X0 6 37 b then there exists on 37 b a unique solution to dny dnil dxn an71x y dxnil g 30Xy hX7 that satisfies the initial conditions d dnil YX0 Ym diX0Y177WIXoni1 0 Example Does the initial value problem y4 7X3y l 3y 0 withy017 yO17 y 0 07 y30 0 have a unique solution on the interval 711 General facts concerning linear differential equations 0 The general solution y to a non homogeneous linear equation of order n is of the form yX yhX YPX7 where yhx is the general solution to the corresponding homogeneous equation and ypx is a particular solution to the non homogeneous equation a The general solution of a homogeneous linear equation of order n with continuous coefficients is a linear combination of n linearly independent solutions General method to solve a linear differential equation As a consequence solving a linear differential equation of order n with continuous coefficients will involve the following steps 0 Find n linearly independent solutions to the homogeneous equation 9 Use the above to write the general solution yhx to the homogeneous equation 9 Find a particular solution ypx to the non homogeneous equation 9 Use the above to write the general solution ygx yhx l ypx to the linear non homogeneous equation 9 Impose the boundary or initial conditions if any hat we will do 0 We will learn methods to c Find linearly independent solutions to o Homogeneous equations with constant coefficients Homogeneous CauchyEuler equations c Find linearly independent solutions to more general homogeneous linear equations assuming we already know one solution 0 Decide whether the solutions are linearly independent 0 Find a particular solution to a nonhomogeneous linear equation 0 We will work mostly with second order differential equations but the methods we will learn can easily be generalized to equations of higher order

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