4.2062   y′′(x)2 = a + by(x)

ODE

y′′(x)2 = a+ by(x)

ODE Classification

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

Book solution method
TO DO

Mathematica
cpu = 0.850654 (sec), leaf count = 119

(     ⌊              (                  )                ⌋      ⌊              (                )                 ⌋)
{       (a + by(x))22F1 1, 2; 5;− 4(a+by(x))3∕2 2                      (a + by(x))22F1 1, 2; 5; 4(a+by(x))3∕2 2              }
 Solve⌈ --------------2-32-3------3bc1-------= (c2 + x)2,y(x)⌉ ,Solve⌈ --------------223-3-----3bc1------= (c2 + x)2,y(x)⌉
(                      b c1                                                     b c1                               )

Maple
cpu = 0.441 (sec), leaf count = 173

(                                                                                                                                                                                                                                   )
||∫                                                         ∫                                                  ∫                                               ∫                                                                     ||
{  y(x)√ -----------------1----------------                  y(x)    ------------b-------------                  y(x) -------------b------------                  y(x)   √ -----------------1----------------                        a-}
||      b 3∘ -(---√-----------√------------)d-a− x − C2 = 0,      − 3∘ ----(--------3∕2--------)d-a− x−  C2 = 0,     3∘ -----(-------3∕2-------)d-a− x − C2 = 0,      − b 3∘ -(---√-----------√------------)d-a− x − C2 = 0,y (x&
(           b 4-a b-a+ ab+ 4a  b a + a−-C1                            − 12b (b-a+ a)   − C1 ∕4                         − 12b (b a+ a)  − -C1∕4                              b 4-a b-a+ ab+ 4a  b a + a−-C1                           )

Mathematica raw input

DSolve[y''[x]^2 == a + b*y[x],y[x],x]

Mathematica raw output

{Solve[(Hypergeometric2F1[1/2, 2/3, 5/3, (-4*(a + b*y[x])^(3/2))/(3*b*C[1])]^2*(
a + b*y[x])^2)/(b^2*C[1]) == (x + C[2])^2, y[x]], Solve[(Hypergeometric2F1[1/2, 
2/3, 5/3, (4*(a + b*y[x])^(3/2))/(3*b*C[1])]^2*(a + b*y[x])^2)/(b^2*C[1]) == (x 
+ C[2])^2, y[x]]}

Maple raw input

dsolve(diff(diff(y(x),x),x)^2 = a+b*y(x), y(x),'implicit')

Maple raw output

y(x) = -a/b, Intat(b*3^(1/2)/(b*(4*_a*(_a*b+a)^(1/2)*b+4*a*(_a*b+a)^(1/2)-_C1))^
(1/2),_a = y(x))-x-_C2 = 0, Intat(-b*3^(1/2)/(b*(4*_a*(_a*b+a)^(1/2)*b+4*a*(_a*b
+a)^(1/2)-_C1))^(1/2),_a = y(x))-x-_C2 = 0, Intat(-3*b/(-12*b*((_a*b+a)^(3/2)-1/
4*_C1))^(1/2),_a = y(x))-x-_C2 = 0, Intat(3*b/(-12*b*((_a*b+a)^(3/2)-1/4*_C1))^(
1/2),_a = y(x))-x-_C2 = 0