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## Main Question or Discussion Point

I am familiar with how to solve a second order, non-homogenous DE with constants, i.e.

[tex]\frac {\partial^2X(t)}{\partial t^2} + \frac{\partial X(t)}{\partial t} = C[/tex]

by first solving the homogenous eqn, then setting the equation equal to a constant, yielding a sol'n of

[tex]X(t)= Ae^{0}+ Be^{-t}+ C[/tex]

But how does one solve a 2nd order equation that also has another t variable in it, such as:

[tex]\frac {\partial^2X(t)}{\partial t^2} + \frac{1}{t} \frac{\partial X(t)}{\partial t} = C[/tex]?

[tex]\frac {\partial^2X(t)}{\partial t^2} + \frac{\partial X(t)}{\partial t} = C[/tex]

by first solving the homogenous eqn, then setting the equation equal to a constant, yielding a sol'n of

[tex]X(t)= Ae^{0}+ Be^{-t}+ C[/tex]

But how does one solve a 2nd order equation that also has another t variable in it, such as:

[tex]\frac {\partial^2X(t)}{\partial t^2} + \frac{1}{t} \frac{\partial X(t)}{\partial t} = C[/tex]?