(************************************************)
(* Regular perturbation: y'' = Sin(x) y;
   y(0) = 1;  y'(0) = 1                        *)
(************************************************)

Remove["Global`*"]
s0 = DSolve[{p''[x]+p[x]==0,p[0]==1,p'[0]==0},p[x],x][[1]];
y0 = Evaluate[p[x]/.s0]

s1 = DSolve[{q''[x]+q[x]==x y0,q[0]==0,q'[0]==0},q[x],x][[1]];
y1 = q[x]/.s1

s2 = DSolve[{r''[x]+r[x]==x y1,r[0]==0,r'[0]==0},r[x],x][[1]];
y2 = r[x]/.s2

s3 = DSolve[{t''[x]+t[x]==x y2,t[0]==0,t'[0]==0},t[x],x][[1]];
y3 = t[x]/.s3

ypert = y0 + y1 + y2 + y3

Remove["Global`*"]
s=DSolve[{y''[x]+(1-ϵ x)y[x]== 0,y[0]==1,y'[0]==0},y[x],x]

ϵ=0.01;
y0 = Cos[x];
y1 = (1/8) (2 x Cos[x] Cos[2 x]-Sin[x]+2 (x^2) Sin[x]+Cos[2 x] Sin[x]-Cos[x] Sin[2 x]+2 x Sin[x] Sin[2 x]);
y2 = (1/192) (21 Cos[x]+12 (x^2) Cos[x]-6 (x^4) Cos[x]-21 Cos[x] Cos[2 x]+30 (x^2) Cos[x] Cos[2 x]+8 (x^3) Sin[x]+42 x Cos[2 x] Sin[x]-12 (x^3) Cos[2 x] Sin[x]-42 x Cos[x] Sin[2 x]+12 (x^3) Cos[x] Sin[2 x]-21 Sin[x] Sin[2 x]+30 (x^2) Sin[x] Sin[2 x]);
y3 =(1/768) (28 (x^3) Cos[x]-8 (x^5) Cos[x]-210 x Cos[x] Cos[2 x]+112 (x^3) Cos[x] Cos[2 x]-6 (x^5) Cos[x] Cos[2 x]+105 Sin[x]+21 (x^4) Sin[x]-2 (x^6) Sin[x]-105 Cos[2 x] Sin[x]+210 (x^2) Cos[2 x] Sin[x]-35 (x^4) Cos[2 x] Sin[x]+105 Cos[x] Sin[2 x]-210 (x^2) Cos[x] Sin[2 x]+35 (x^4) Cos[x] Sin[2 x]-210 x Sin[x] Sin[2 x]+112 (x^3) Sin[x] Sin[2 x]-6 (x^5) Sin[x] Sin[2 x]);
ypert = y0 + ϵ y1 + (ϵ^2) y2 +(ϵ^3) y3;

Plot[{((AiryAiPrime[-(1/(ϵ^(2/3)))] AiryBi[((-1+x ϵ)/(ϵ^(2/3)))]-AiryAi[((-1+x ϵ)/(ϵ^(2/3)))] AiryBiPrime[-(1/(ϵ^(2/3)))])/(AiryAiPrime[-(1/(ϵ^(2/3)))] AiryBi[-(1/(ϵ^(2/3)))]-AiryAi[-(1/(ϵ^(2/3)))] AiryBiPrime[-(1/(ϵ^(2/3)))])),ypert},{x,0,30},PlotStyle->Automatic]