Example sinx and (linearly independent) e*and ex(linearly independent) sin2x and sinxcosx (linearly dependent) and 4口0y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
Example sinx and Icosx (linearly independent) e*and ex(linearly independent) sin2x and sinxcosx (linearly dependent) x and (linearly independent on the entire real line 4口卡0y至，无，3000
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
Example sinx and cosx (linearly independent) e*and ex (linearly independent) sin2x and sinxcosx (linearly dependent) x and (linearly independent on the entire real line but linearly dependent on the interval(0,)) 4口0y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
Example sinx and cosx (linearly independent) e*and ex(linearly independent) sin2x and sinxcosx (linearly dependent) x andx(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
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
General solutions Question 1:Does y”+p(x)y+q(x)y=0 always have two linearly independent solutions? 4口0y至，无2000
General solutions Question 1: Does y 00 +p(x)y 0 +q(x)y = 0 always have two linearly independent solutions? (Yes!) Question 2: Given any two linearly independent solutions y1 and y2 of y 00 +p(x)y 0 +q(x)y = 0 (10) Does every solution of (10) can be expressed as y = c1y1 +c2y2? (Yes!)