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 the solution of the given differential equation.

Step-by-step solution

Given Information

The given value is

and

To differentiate the given power series

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

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

And the second derivative

Use the Substitution method.

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

Solve the expression.

If the above equation is reduced to zero, this implies that the given power series is a solution to the given differential equation. Now, is to make the summation index and the power offor the three series, in the same phase. We, first, have to assure that the first terms, for the three-power series, are raised to the same power, which is the case. Now, we do the shift by replacing the summation index, for the third power series, byinstead of n. Note that, this wouldn't affect the terms of the series, The shift yields the same terms.

For the first series, we letThe summation becomes:

Thus, the given power series is the solution of the given differential equation.