Suggestions.
(1) Place short code questions in the question rather than an attached notebook.
(2) Avoid upper case for function and variable names (less likely to get confused for built-in symbols).
(3) Keep the code as simple as possible-- avoid Subscript
when it is not needed for example.
(4) Explain what results you got and why they are not doing what you want.
Here are the examples.
DSolve[{f'[x] + \[Lambda]^2*f[x] == 0 }, {f[x]}, x]
(* Out[2]= {{f[x] -> E^(-x \[Lambda]^2) C[1]}} *)
DSolve[{g''[y] + (\[Epsilon]*(y^2 a1 + y*a2 + a3))/
a4*\[Lambda]^2*g[y] == 0 }, {g[y]}, y]
(* Out[3]= {{g[y] ->
C[1] ParabolicCylinderD[(-4 a1^(3/2) Sqrt[a4] +
I a2^2 Sqrt[\[Epsilon]] \[Lambda] -
4 I a1 a3 Sqrt[\[Epsilon]] \[Lambda])/(
8 a1^(3/2) Sqrt[
a4]), ((-1)^(1/4) a2 \[Epsilon]^(1/4) Sqrt[\[Lambda]])/(
Sqrt[2] a1^(3/4) a4^(1/4)) + ((-1)^(1/4) Sqrt[2] a1^(1/4)
y \[Epsilon]^(1/4) Sqrt[\[Lambda]])/a4^(1/4)] +
C[2] ParabolicCylinderD[(-4 a1^(3/2) Sqrt[a4] -
I a2^2 Sqrt[\[Epsilon]] \[Lambda] +
4 I a1 a3 Sqrt[\[Epsilon]] \[Lambda])/(8 a1^(3/2) Sqrt[a4]),
I (((-1)^(1/4) a2 \[Epsilon]^(1/4) Sqrt[\[Lambda]])/(
Sqrt[2] a1^(3/4) a4^(1/4)) + ((-1)^(1/4) Sqrt[2] a1^(1/4)
y \[Epsilon]^(1/4) Sqrt[\[Lambda]])/a4^(1/4))]}} *)
It is not clear from the post what were the desired results.