ODE
\[ y''(x)^2=a+b y(x) \] 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
\[\left \{\text {Solve}\left [\frac {(a+b y(x))^2 \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 c_1}=\left (c_2+x\right ){}^2,y(x)\right ],\text {Solve}\left [\frac {(a+b y(x))^2 \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 c_1}=\left (c_2+x\right ){}^2,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.441 (sec), leaf count = 173
\[ \left \{ \int ^{y \left ( x \right ) }\!{b\sqrt {3}{\frac {1}{\sqrt {b \left ( 4\,{\it \_a}\,\sqrt {b{\it \_a}+a}b+4\,a\sqrt {b{\it \_a}+a}-{\it \_C1} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-3\,{\frac {b}{\sqrt {-12\,b \left ( \left ( b{\it \_a}+a \right ) ^{3/2}-{\it \_C1}/4 \right ) }}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!3\,{\frac {b}{\sqrt {-12\,b \left ( \left ( b{\it \_a}+a \right ) ^{3/2}-{\it \_C1}/4 \right ) }}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{b\sqrt {3}{\frac {1}{\sqrt {b \left ( 4\,{\it \_a}\,\sqrt {b{\it \_a}+a}b+4\,a\sqrt {b{\it \_a}+a}-{\it \_C1} \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,y \left ( x \right ) =-{\frac {a}{b}} \right \} \] 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