Linear ode with non-constant coeﬃcients A(x) y′′ + B(x) y′ + C(x) y = f(x)

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4.3.2 Linear ode with non-constant coeﬃcients \(A\left ( x\right ) y^{\prime \prime }+B\left ( x\right ) y^{\prime }+C\left ( x\right ) y=f\left ( x\right ) \)

Euler ode \(x^{2}y^{\prime \prime }+xy^{\prime }+y=f\left ( x\right ) \)
Kovacic type
Method of conversion to ﬁrst order Riccati
Airy ode \(y^{\prime \prime }\pm kxy=0\) or \(y^{\prime \prime }+by^{\prime }\pm kxy=0\)
Solved using series method
Reduction of order
Transformation to a constant coeﬃcient ODE methods
Exact linear second order ode \(p\left ( x\right ) y^{\prime \prime }+q\left ( x\right ) y^{\prime }+r\left ( x\right ) y=0\)
Linear second order not exact but solved by ﬁnding mu(x) integrating factor.
Linear second order not exact but solved by ﬁnding an M integrating factor.
Solved using Lagrange adjoint equation method.
Solved By transformation on \(B\left ( x\right ) \) for ODE \(Ay^{\prime \prime }\left ( x\right ) +By^{\prime }\left ( x\right ) +C\left ( x\right ) y\left ( x\right ) =0\)
Bessel type ode \(x^{2}y^{\prime \prime }+xy^{\prime }+\left ( x^{2}-n^{2}\right ) y=f\left ( x\right ) \)
Bessel form A type ode \(ay^{\prime \prime }+by^{\prime }+(ce^{rx}-m)y=f\left ( x\right ) \)

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