4.37.10 \(a y'(x)^2+b y(x)+y''(x)=0\)

ODE
\[ a y'(x)^2+b y(x)+y''(x)=0 \] ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.804371 (sec), leaf count = 104

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[1]} c_1 a^2-2 b K[1] a+b}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {2} a}{\sqrt {2 e^{-2 a K[2]} c_1 a^2-2 b K[2] a+b}}dK[2]\& \right ][x+c_2]\right \}\right \}\]

Maple ✓
cpu = 0.779 (sec), leaf count = 79

\[\left [\int _{}^{y \left (x \right )}-\frac {2 a}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \,a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}\frac {2 a}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} \textit {\_C1} \,a^{2}-4 \textit {\_a} a b +2 b}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input

DSolve[b*y[x] + a*y'[x]^2 + y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[Inactive[Integrate][-((Sqrt[2]*a)/Sqrt[b + (2*a^2*C[1]
)/E^(2*a*K[1]) - 2*a*b*K[1]]), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFu
nction[Inactive[Integrate][(Sqrt[2]*a)/Sqrt[b + (2*a^2*C[1])/E^(2*a*K[2]) - 2*a*
b*K[2]], {K[2], 1, #1}] & ][x + C[2]]}}

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*y(x) = 0, y(x))

Maple raw output

[Intat(-2/(4*exp(-2*a*_a)*_C1*a^2-4*_a*a*b+2*b)^(1/2)*a,_a = y(x))-x-_C2 = 0, In
tat(2/(4*exp(-2*a*_a)*_C1*a^2-4*_a*a*b+2*b)^(1/2)*a,_a = y(x))-x-_C2 = 0]