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 \(A\left ( x\right ) y^{\prime \prime }+B\left ( x\right ) y^{\prime }+C\left ( x\right ) y=f\left ( x\right ) \)

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

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