In Problem proceed as in Example 4 and find a power series solution of the given linear first order differential equation.

(reference example 4)

Solution. Our aim is to find the general solution of the given differential equation using power series definition.

Given

……………. (1)

Given differential equation is

............................................ (2)

Step 1. References are given below.

Reference 1.

............. ref.(1)

Step 2.

Differentiate w.r.t x then we get

Put this value in the given differential equation 2. then we get

Put

It is an identical so we must have

for all k=0,1,2,3,4,5,.......................................... (3)