((1+t^2)y')'-(t^2-1)y=t oscillatory or nonoscillatory?
And Why it is oscillatory or nonoscillatoy?
When you solve the equation, what kind of elementary functions form your solution?
If it's periodic functions like sine and cosine, then the motion described by the equation is oscillatory.
Otherwise, if it's exponential functions, then the motion described by the equation is non-oscillatory.