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[tex](a+bx+cx^2)y^{''}+(f+gx+hx^2)y^{'}+(j+kx+mx^2)y=0[/tex]

Does anyone here know if it's been "completely" characterized in terms of the geometry of the three parabolas which make up it's coefficients?

For example, if I'm given plots of the parabolas, can any information at all be extracted from them in order to determine at the very least the general appearance of the solution of the corresponding DE without having to directly solve it?

No doubt someone can just start intensively investigating the solutions directly but I suppose that's already been done. Anyone know about this?

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# Characterizing second order ODE

Can you offer guidance or do you also need help?

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