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 $\"\"$ $\"\"$

$\"\"$ is a recursive relation.

Find coefficients for some values of $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Substitute above values in $\"\"$

$\"\"$

From the above expression 3 multiples of coefficients only remained

we can write $\"\"$ .

Rewrite the sum as $\"\"$ .

Maclaurin series is $\"\"$ .

write the sum in the exponential form

$\"\"$

Solution:

Solution of the differential equation $\"\"$ is $\"\"$