Question

Use power series to solve the differential equation.

Answer 1

Step 1:

The differential equation is .

Assume there is a solution of the power series form

Determine the derivative of the above solution function with respect to .

Substitute in .

Change to in sigma notation

The above expression is zero when and coefficient of is zero.

This will result , and

Step 2:

is a recursive relation.

Find coefficients for some values of .

Substitute above values in

From the above expression, the coefficient repeat at multiples of n = 3.

we can write .

Rewrite the sum as .

Maclaurin series is .

Write the sum in the exponential form.

Solution:

Solution of the differential equation is .