ODE
\[ y(x) y''(x)=a y'(x)^2+b \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.613567 (sec), leaf count = 206
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \, _2F_1\left (\frac {1}{2},\frac {1}{2 a};1+\frac {1}{2 a};\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {\text {$\#$1}^{2 a} e^{2 a c_1}-b}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {a} \sqrt {\frac {b-\text {$\#$1}^{2 a} e^{2 a c_1}}{b}} \, _2F_1\left (\frac {1}{2},\frac {1}{2 a};1+\frac {1}{2 a};\frac {e^{2 a c_1} \text {$\#$1}^{2 a}}{b}\right )}{\sqrt {\text {$\#$1}^{2 a} e^{2 a c_1}-b}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.162 (sec), leaf count = 64
\[ \left \{ \int ^{y \left ( x \right ) }\!{a{\frac {1}{\sqrt {-a \left ( -{\it \_C1}\,{{\it \_a}}^{2\,a}+b \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{a{\frac {1}{\sqrt {-a \left ( -{\it \_C1}\,{{\it \_a}}^{2\,a}+b \right ) }}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == b + a*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((Sqrt[a]*Hypergeometric2F1[1/2, 1/(2*a), 1 + 1/(2*a), (E^(2*a*C[1])*#1^(2*a))/b]*#1*Sqrt[(b - E^(2*a*C[1])*#1^(2*a))/b])/Sqrt[-b + E^(2*a*C[1])*#1^(2*a)]) & ][x + C[2]]}, {y[x] -> InverseFunction[(Sqrt[a]*Hypergeometric2F1[1/2, 1/(2*a), 1 + 1/(2*a), (E^(2*a*C[1])*#1^(2*a))/b]*#1*Sqrt[(b - E^(2*a*C[1])*#1^(2*a))/b])/Sqrt[-b + E^(2*a*C[1])*#1^(2*a)] & ][x + C[2]]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a*diff(y(x),x)^2+b, y(x),'implicit')
Maple raw output
Intat(a/(-a*(-_C1*_a^(2*a)+b))^(1/2),_a = y(x))-x-_C2 = 0, Intat(-a/(-a*(-_C1*_a^(2*a)+b))^(1/2),_a = y(x))-x-_C2 = 0