Is there an easier way to solve this problem than just trial & error?

Match the solutions to the differential equations.

1. (x^2)y''+(6x)y' -6y = 0
2. (x^2)y''-(2x)y' -4y = 0
3. y''-25y' = 0

A. y = x^7
B. y=x^4 or y =x^-3
C. y = x^-6
D. y = x^4 or y = x^-1
E. y = e^4x
F. y = e^5x or y = e^-5x

Question

Is there an easier way to solve this problem than just trial & error?

Match the solutions to the differential equations. 

1. (x^2)y''+(6x)y' -6y = 0
2. (x^2)y''-(2x)y' -4y = 0
3. y''-25y' = 0

A. y = x^7
B. y=x^4 or y =x^-3
C. y = x^-6
D. y = x^4 or y = x^-1
E. y = e^4x
F. y = e^5x or y = e^-5x

turingtest · Answer

you could easily solve the third, and the others can be done with Laplace transforms. There are probably a few other ways too, but trial ans error will probably be best for the first three DE's if you don't know Laplace... or maybe even if you do for that matter.