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Chapter 3
Linear second order ode
3.1
Quadrature ode
\(y^{\prime \prime }=f\left ( x\right ) \)
3.2
Linear ode with constant coefficients
\(Ay^{\prime \prime }+By^{\prime }+Cy=f\left ( x\right ) \)
3.3
Linear ode with non-constant coefficients
\(A\left ( x\right ) y^{\prime \prime }+B\left ( x\right ) y^{\prime }+C\left ( x\right ) y=f\left ( x\right ) \)
3.4
Collection of special transformations
3.5
Euler ode
\(x^{2}y^{\prime \prime }+xy^{\prime }+y=f\left ( x\right ) \)
3.6
Method of conversion to first order Riccati
3.7
Airy ode
\(y^{\prime \prime }\pm kxy=0\)
or
\(y^{\prime \prime }+by^{\prime }\pm kxy=0\)
3.8
Reduction of order
3.9
Transformation to a constant coefficient ODE methods
3.10
Exact linear second order ode
\(p_{2}\left ( x\right ) y^{\prime \prime }+p_{1}\left ( x\right ) y^{\prime }+p_{0}\left ( x\right ) y=f\left ( x\right ) \)
3.11
Linear second order not exact but solved by finding an integrating factor.
3.12
Linear second order not exact but solved by finding an M integrating factor.
3.13
Solved using Lagrange adjoint equation method.
3.14
Solved By transformation on
\(B\left ( x\right ) \)
for ODE
\(Ay^{\prime \prime }\left ( x\right ) +By^{\prime }\left ( x\right ) +C\left ( x\right ) y\left ( x\right ) =0\)
3.15
Bessel type ode
\(x^{2}y^{\prime \prime }+xy^{\prime }+\left ( x^{2}-n^{2}\right ) y=f\left ( x\right ) \)
3.16
Bessel form A type ode
\(ay^{\prime \prime }+by^{\prime }+(ce^{rx}-m)y=f\left ( x\right ) \)
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