How Does the Wronskian Affect Linear Independence in Second Order ODEs?

[mathman44]

y1 and y2 are solutions to the ODE

$$L[y]=0=y''+p(x)y'+q(x)y$$What can you conclude about p(x), q(x) and the solutions on the interval I

if

i) $$W(x) = 0$$ for all X on I
ii) $$W(x) = c$$ for all X on I, c =/= 0

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$$W(x) = y_1'y_2-y_1y_2' = C*e^{\int{p(x)}}$$

i) W=0 so y1'y2=y1y2'

And y1 and y2 are not linearly independent.

Not sure what else I can conclude here.

ii) By Abel's theorem, p(x) = 0

y1 and y2 are linearly independent.
What am I missing?

 

[mighty2000]

For part i), you can also conclude that the solutions y1 and y2 are constant multiples of each other. This is because if y1'y2=y1y2', then y1'/y1 = y2'/y2, meaning that the ratio of the derivatives is constant. Therefore, y1 and y2 are related by a constant factor.

For part ii), you can also conclude that p(x) = 0 for all x on I, not just for a specific c. This is because if W(x) = c for all x on I, then by Abel's theorem, p(x) must be constant on I. But since we know that W(x) = c for all x on I, not just for a specific c, then p(x) must be equal to 0 for all x on I.