ODE
\[ y'(x)^2 (\text {a0}+\text {b0} x+\text {c0} y(x))+y'(x) (\text {a1}+\text {b1} x+\text {c1} y(x))+\text {a2}+\text {b2} x+\text {c2} y(x)=0 \] ODE Classification
[_dAlembert]
Book solution method
Change of variable
Mathematica ✗
cpu = 600. (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 0.168 (sec), leaf count = 477
\[ \left \{ [x \left ( {\it \_T} \right ) ={{\rm e}^{\int \!{\frac { \left ( -{\it b0}\,{\it c1}+{\it b1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( -2\,{\it b0}\,{\it c2}+2\,{\it b2}\,{\it c0} \right ) {\it \_T}-{\it b1}\,{\it c2}+{\it b2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }}\,{\rm d}{\it \_T}}} \left ( \int \!-{\frac { \left ( {\it a0}\,{\it c1}-{\it a1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( 2\,{\it a0}\,{\it c2}-2\,{\it a2}\,{\it c0} \right ) {\it \_T}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }{{\rm e}^{-\int \!{\frac { \left ( -{\it b0}\,{\it c1}+{\it b1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( -2\,{\it b0}\,{\it c2}+2\,{\it b2}\,{\it c0} \right ) {\it \_T}-{\it b1}\,{\it c2}+{\it b2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }}\,{\rm d}{\it \_T}}}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) ,y \left ( {\it \_T} \right ) =-{\frac {{{\it \_T}}^{2}{\it b0}+{\it \_T}\,{\it b1}+{\it b2}}{{{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2}}{{\rm e}^{\int \!{\frac { \left ( -{\it b0}\,{\it c1}+{\it b1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( -2\,{\it b0}\,{\it c2}+2\,{\it b2}\,{\it c0} \right ) {\it \_T}-{\it b1}\,{\it c2}+{\it b2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }}\,{\rm d}{\it \_T}}} \left ( \int \!-{\frac { \left ( {\it a0}\,{\it c1}-{\it a1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( 2\,{\it a0}\,{\it c2}-2\,{\it a2}\,{\it c0} \right ) {\it \_T}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }{{\rm e}^{-\int \!{\frac { \left ( -{\it b0}\,{\it c1}+{\it b1}\,{\it c0} \right ) {{\it \_T}}^{2}+ \left ( -2\,{\it b0}\,{\it c2}+2\,{\it b2}\,{\it c0} \right ) {\it \_T}-{\it b1}\,{\it c2}+{\it b2}\,{\it c1}}{ \left ( {{\it \_T}}^{3}{\it c0}+ \left ( {\it b0}+{\it c1} \right ) {{\it \_T}}^{2}+ \left ( {\it b1}+{\it c2} \right ) {\it \_T}+{\it b2} \right ) \left ( {{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2} \right ) }}\,{\rm d}{\it \_T}}}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) }+{\frac {-{{\it \_T}}^{2}{\it a0}-{\it \_T}\,{\it a1}-{\it a2}}{{{\it \_T}}^{2}{\it c0}+{\it \_T}\,{\it c1}+{\it c2}}}] \right \} \] Mathematica raw input
DSolve[a2 + b2*x + c2*y[x] + (a1 + b1*x + c1*y[x])*y'[x] + (a0 + b0*x + c0*y[x])*y'[x]^2 == 0,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve((a0+b0*x+c0*y(x))*diff(y(x),x)^2+(a1+b1*x+c1*y(x))*diff(y(x),x)+a2+b2*x+c2*y(x) = 0, y(x),'implicit')
Maple raw output
[x(_T) = exp(Int(((-b0*c1+b1*c0)*_T^2+(-2*b0*c2+2*b2*c0)*_T-b1*c2+b2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T))*(Int(-exp(-Int(((-b0*c1+b1*c0)*_T^2+(-2*b0*c2+2*b2*c0)*_T-b1*c2+b2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T))*((a0*c1-a1*c0)*_T^2+(2*a0*c2-2*a2*c0)*_T+a1*c2-a2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T)+_C1), y(_T) = -(_T^2*b0+_T*b1+b2)/(_T^2*c0+_T*c1+c2)*exp(Int(((-b0*c1+b1*c0)*_T^2+(-2*b0*c2+2*b2*c0)*_T-b1*c2+b2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T))*(Int(-exp(-Int(((-b0*c1+b1*c0)*_T^2+(-2*b0*c2+2*b2*c0)*_T-b1*c2+b2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T))*((a0*c1-a1*c0)*_T^2+(2*a0*c2-2*a2*c0)*_T+a1*c2-a2*c1)/(_T^3*c0+(b0+c1)*_T^2+(b1+c2)*_T+b2)/(_T^2*c0+_T*c1+c2),_T)+_C1)+(-_T^2*a0-_T*a1-a2)/(_T^2*c0+_T*c1+c2)]