Analytic Methods for Solving Second-Order ODE

The general form of a second-order ode is

Usually, the differential equation is accompanied by initial conditions
or boundary conditions.

Classes of Second-Order ODE that can be Solved Analytically

It is said that a differential equation is solved exactly if the answer can
be expressed in the form of an integral. It may be possible to solve the
following classes of second-order ode exactly:

• linear

  

• reducible (class 1)

  

  The second derivative depends on the independent variable t
  and the first derivative only.
  

• reducible (class 2)

  
  
  The second derivative depends on the dependent variable and its
  derivative only.
  

If a second-order ode cannot be solved analytically, then it can always be
approximated by a numerical method.

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