To find the unique solution to the IVP Jy"+p(x)y+q(x)y=0, y(a)=bo,y(a)=b1. Firstly,we find two essentially different solutions y and y2; Secondly,impose the initial conditions y(a)=bo,y(a)=bi on the general solution y(x)=c1y1+c2y2 to get the simultaneous equations c1y1(a）+c2y2(a）=bo, c1M(a)+c22(a）=b1. for the coefficients ci and c2. 4口①，t元2月00
To find the unique solution to the IVP ( y 00 +p(x)y 0 +q(x)y = 0, y(a) = b0, y 0 (a) = b1. Firstly, we find two essentially different solutions y1 and y2; Secondly, impose the initial conditions y(a) = b0, y 0 (a) = b1 on the general solution y(x) = c1y1 +c2y2 to get the simultaneous equations ( c1y1(a) +c2y2(a) = b0, c1y 0 1 (a) +c2y 0 2 (a) = b1. for the coefficients c1 and c2
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on provided that neither is a constant multiple of the other. 1口1日，4元卡2000
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on I provided that neither is a constant multiple of the other. Two functions are said to be linearly dependent on an open interval provided that one of them is a constant multiple of the other. We can always determine whether two given functions f and g are linearly dependent on an interval I by noting at a glance whether either of the two quotients f g or g f a constantvalued function on I
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on provided that neither is a constant multiple of the other. Two functions are said to be linearly dependent on an open interval provided that one of them is a constant multiple of the other. 1口1日，元卡20只0
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on I provided that neither is a constant multiple of the other. Two functions are said to be linearly dependent on an open interval provided that one of them is a constant multiple of the other. We can always determine whether two given functions f and g are linearly dependent on an interval I by noting at a glance whether either of the two quotients f g or g f a constantvalued function on I
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on provided that neither is a constant multiple of the other. Two functions are said to be linearly dependent on an open interval provided that one of them is a constant multiple of the other.We can always determine whether two given functions f and g are linearly dependent on an interval I by noting at a glance whether either of the two quotientsora constant- valued function on / 4口0y至无3000
Linear Independence of Two Functions Definition (Linear Independence of Two Functions) Two functions defined on an open interval I are said to be linearly independent on I provided that neither is a constant multiple of the other. Two functions are said to be linearly dependent on an open interval provided that one of them is a constant multiple of the other. We can always determine whether two given functions f and g are linearly dependent on an interval I by noting at a glance whether either of the two quotients f g or g f a constantvalued function on I
Example sinx and cOSx 4日10y至，无2000
Example sinx and cos x (linearly independent) e x and e −x (linearly independent) sin 2x and sinx cos x (linearly dependent) x and |x| (linearly independent on the entire real line but linearly dependent on the interval (0,+∞)) Remark The identically zero function f(x) ≡ 0 and any other function g is linearly dependent on every interval