Question

In Problems, 31-34 verify by direct substitution that the given power series is a solution of the indicated differential equation. [Hint: For a power

Short answer

Answer:

Therefore, the given power series is a solution to the given differential equation.

Step-by-step solution

Given Information

The given value is

and

Use Differentiation

We aim to check if the given power series is a solution for the given differential equation or not. We find the first derivative of the power series as follows:

We shift the summation index toas the zeroth term of the power series equals to zero.

Use the Substitution method

We substitute (1) and (2) into the given differential equation, yields

If the above equation is reduced to zero, this implies that the given power series is a solution to the given differential equation. We make the summation index and the powers offor the two series, into the same phase by shifting the summation index in the first series toinstead of

Therefore, the given power series is a solution for the given differential equation.