Question

How do you use Power Series to solve the differential equation y'−y=0 ?

Answer 1

The solution is

y=c0∞∑n=0xnn!=c0ex,

where c0 is any constant.

Let us look at some details.

Let 
y=∞∑n=0cnxn
y'=∞∑n=1ncnxn−1=∞∑n=0(n+1)cn+1xn

So, we can rewrite y'−y=0 as

∞∑n=0(n+1)cn+1xn−∞∑n=0cnxn=0

by combining the summations,

⇒∞∑n=0[(n+1)cn+1−cn]xn=0

so, we have

(n+1)cn+1−cn=0⇒cn+1=1n+1cn

Let us observe the first few terms.

c1=11c0=11!c0

c2=12c1=12⋅11!c0=12!c0

c3=13c2=13⋅12!c0=13!c0
.
.
.
cn=1n!c0

Hence, the solution is

y=∞∑n=01n!c0xn=c0∞∑n=0xnn!=c0ex,

where c0 is any constant.