Second Order Linear Differential Equations

16. y ′′ − 6y ′ +9y =0; y1(x) =e3x .
17. y ′′ − 2
x y′ + 2
x2 y =0; y1(x) =x.
18. y ′′ − 1
x y′ + 1
x 2 y =0; y1(x) =x.
19. y ′′ − 1
x y′ − 4x 2 y =0; y1(x) =e x2
.
20. y ′′ 2x − 1
− y
x
′ x − 1
+
x y =0; y1(x)=ex .
21. Let y = y1(x) and y = y2(x) be solutions of equation (H) on an interval I. Let
a ∈ I and suppose that
y1(a) =α, y ′ 1(a) =β and y2(a) =γ, y ′ 2(a) =δ.
Under what conditions on α, β, γ, δ will the functions y1 and y2 be linearly
independent on I?
22. Suppose that the functions y1 and y2 are linearly independent solutions of (H).
Does it follow that c1y1 and c2y2 are also linearly independent solutions of (H)? If
not, why not.
23. Suppose that the functions y1 and y2 are linearly independent solutions of (H).
Prove that y3 = y1 + y2 and y4 = y1 − y2 are also linearly independent solutions of
(H). Conversely, prove that if y3 and y4 are linearly independent solutions of (H),
then y1 and y2 are linearly independent solutions of (H).
24. Suppose that the functions y1 and y2 are linearly independent solutions of (H).
Under what conditions will the functions y3 = αy1 + βy2 and y4 = γy1 + δy2 be
linearly independent solutions of (H)?
25. Suppose that y = y1(x) and y = y2(x) are solutions of (H). Show that if y1(x) = 0
on I and W [y1,y2](x) ≡ 0 on I, then y2(x) =λy1(x) on I.
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