SOLUTION: For this problem you may want to recall that if y=sinh(x), then dy/dx=cosh(x), and d^(2)y/dx^(2)=sinh(x). It can be shown that y1=sinh(3x), y2=cosh(3x) and y3=e^(3x) are solutio

Question 1184638: For this problem you may want to recall that if y=sinh(x), then dy/dx=cosh(x), and d^(2)y/dx^(2)=sinh(x).
It can be shown that y1=sinh(3x), y2=cosh(3x) and y3=e^(3x) are solutions to the differential equation xD^(3)y−8D^(2)y−9xDy+72y=0 on (0,∞).
a)
What does the Wronskian of y1,y2,y3 equal on (0,∞)?
W(y1,y2,y3)=______________________ on (0,∞).
b)
Is {y1,y2,y3} a fundamental set for xD^(3)y−8D^(2)y−9xDy+72y=0 on (0,∞)?
Yes or No
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!

a) .
***This is expected since the set { , , } is linearly dependent, because .

b) No, { , , } is not a fundamental set for the D.E. (i.e., it cannot be a basis for the solutions of the D.E.).
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