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 = 1.06428 (sec), leaf count = 119
\[\left \{\text {Solve}\left [\frac {(a+b y(x))^2 \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 c_1}=(x+c_2){}^2,y(x)\right ],\text {Solve}\left [\frac {(a+b y(x))^2 \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 c_1}=(x+c_2){}^2,y(x)\right ]\right \}\]
Maple ✓
cpu = 4.345 (sec), leaf count = 201
\[\left [y \left (x \right ) = -\frac {a}{b}, \int _{}^{y \left (x \right )}\frac {b \sqrt {3}}{\sqrt {b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 a \sqrt {b \textit {\_a} +a}-\textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {b \sqrt {3}}{\sqrt {b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 a \sqrt {b \textit {\_a} +a}-\textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {3 b}{\sqrt {-3 b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 a \sqrt {b \textit {\_a} +a}-\textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}\frac {3 b}{\sqrt {-3 b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 a \sqrt {b \textit {\_a} +a}-\textit {\_C1} \right )}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] 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))
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/(-3*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/(-3*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]