Finding a Taylor Series for the Solution to a Differential Equation Given Initial Conditions

If the exact solution to a differential equation cannot be found one method of solving the equation is by repeated differentiation to find a Taylor series for the solution. In general the series will involve the introduction of constants, but these can be found if we have some initial or boundary conditions for the solution.

For example, ifwiththen we can find the first two terms in the Taylor series

using the differential equation itself. We are givenand can rearrange the equation to giveSubstituteandto get

then

We can get the third and fourth terms by differentiating the differential equation to give

Substituteandto get

Now differentiateto get

Substituteandto get

Then