What is the particular solution of this differential equation 5y'''-y''-6y'=1+x^2

Reduce the order of the ODE by setting and .



Consider the homogeneous ODE



which has characteristic equation



which has roots at and , so that the characteristic solution is



For the nonhomogeneous ODE,



we can expect a particular solution of the form





Substituting these expressions into the ODE yields




from which it follows that



and so the particular solution is



and the general solution for is




Integrate both sides once to solve for :

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