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
[_rational, _dAlembert]
Book solution method
Change of variable
Mathematica ✓
cpu = 8.08068 (sec), leaf count = 576
\[\text {Solve}\left [\left \{x=-\frac {-(K[2] (\text {c0} K[2]+\text {c1})+\text {c2}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {c0}+\text {$\#$1}^2 \text {b0}+\text {$\#$1}^2 \text {c1}+\text {$\#$1} \text {b1}+\text {$\#$1} \text {c2}+\text {b2}\& ,\frac {\text {$\#$1}^2 \text {c0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {c1} \log (K[2]-\text {$\#$1})+\text {c2} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {c0}+2 \text {$\#$1} \text {b0}+2 \text {$\#$1} \text {c1}+\text {b1}+\text {c2}}\& \right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {c0} \text {$\#$1}^3+\text {b0} \text {$\#$1}^2+\text {c1} \text {$\#$1}^2+\text {b1} \text {$\#$1}+\text {c2} \text {$\#$1}+\text {b2}\& ,\frac {\text {c0} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {c1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {c2} \log (K[1]-\text {$\#$1})}{3 \text {c0} \text {$\#$1}^2+2 \text {b0} \text {$\#$1}+2 \text {c1} \text {$\#$1}+\text {b1}+\text {c2}}\& \right ]\right ) (-\text {a2}-K[1] (\text {a1}+\text {a0} K[1]))}{\text {b2}+K[1] (\text {b1}+\text {c2}+K[1] (\text {b0}+\text {c1}+\text {c0} K[1]))}dK[1]+c_1\right )+\text {a0} K[2]^2+\text {a1} K[2]+\text {a2}}{K[2] (K[2] (\text {c0} K[2]+\text {b0}+\text {c1})+\text {b1}+\text {c2})+\text {b2}},y(x)=-\frac {K[2] (K[2] (\text {a0} K[2]+\text {a1})+\text {a2})+(K[2] (\text {b0} K[2]+\text {b1})+\text {b2}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {c0}+\text {$\#$1}^2 \text {b0}+\text {$\#$1}^2 \text {c1}+\text {$\#$1} \text {b1}+\text {$\#$1} \text {c2}+\text {b2}\& ,\frac {\text {$\#$1}^2 \text {c0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {c1} \log (K[2]-\text {$\#$1})+\text {c2} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {c0}+2 \text {$\#$1} \text {b0}+2 \text {$\#$1} \text {c1}+\text {b1}+\text {c2}}\& \right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {c0} \text {$\#$1}^3+\text {b0} \text {$\#$1}^2+\text {c1} \text {$\#$1}^2+\text {b1} \text {$\#$1}+\text {c2} \text {$\#$1}+\text {b2}\& ,\frac {\text {c0} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {c1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {c2} \log (K[1]-\text {$\#$1})}{3 \text {c0} \text {$\#$1}^2+2 \text {b0} \text {$\#$1}+2 \text {c1} \text {$\#$1}+\text {b1}+\text {c2}}\& \right ]\right ) (-\text {a2}-K[1] (\text {a1}+\text {a0} K[1]))}{\text {b2}+K[1] (\text {b1}+\text {c2}+K[1] (\text {b0}+\text {c1}+\text {c0} K[1]))}dK[1]+c_1\right )}{K[2] (K[2] (\text {c0} K[2]+\text {b0}+\text {c1})+\text {b1}+\text {c2})+\text {b2}}\right \},\{y(x),K[2]\}\right ]\]
Maple ✓
cpu = 2.688 (sec), leaf count = 933
\[\left [x -{\mathrm e}^{\int _{}^{-\frac {\mathit {b1} x +\mathit {c1} y \left (x \right )+\sqrt {-4 \mathit {b0} \mathit {b2} \,x^{2}-4 \mathit {b0} \mathit {c2} x y \left (x \right )+\mathit {b1}^{2} x^{2}+2 \mathit {b1} \mathit {c1} x y \left (x \right )-4 \mathit {b2} \mathit {c0} x y \left (x \right )-4 \mathit {c0} \mathit {c2} y \left (x \right )^{2}+\mathit {c1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {b2} x -4 \mathit {a0} \mathit {c2} y \left (x \right )+2 \mathit {a1} \mathit {b1} x +2 \mathit {a1} \mathit {c1} y \left (x \right )-4 \mathit {a2} \mathit {b0} x -4 \mathit {a2} \mathit {c0} y \left (x \right )-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}+\mathit {a1}}{2 \left (\mathit {a0} +\mathit {b0} x +\mathit {c0} y \left (x \right )\right )}}-\frac {\textit {\_a}^{2} \mathit {b0} \mathit {c1} -\textit {\_a}^{2} \mathit {b1} \mathit {c0} +2 \textit {\_a} \mathit {b0} \mathit {c2} -2 \textit {\_a} \mathit {b2} \mathit {c0} +\mathit {b1} \mathit {c2} -\mathit {b2} \mathit {c1}}{\left (\textit {\_a}^{3} \mathit {c0} +\textit {\_a}^{2} \mathit {b0} +\textit {\_a}^{2} \mathit {c1} +\textit {\_a} \mathit {b1} +\textit {\_a} \mathit {c2} +\mathit {b2} \right ) \left (\textit {\_a}^{2} \mathit {c0} +\mathit {c1} \textit {\_a} +\mathit {c2} \right )}d \textit {\_a}} \left (\int _{}^{-\frac {\mathit {b1} x +\mathit {c1} y \left (x \right )+\sqrt {-4 \mathit {b0} \mathit {b2} \,x^{2}-4 \mathit {b0} \mathit {c2} x y \left (x \right )+\mathit {b1}^{2} x^{2}+2 \mathit {b1} \mathit {c1} x y \left (x \right )-4 \mathit {b2} \mathit {c0} x y \left (x \right )-4 \mathit {c0} \mathit {c2} y \left (x \right )^{2}+\mathit {c1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {b2} x -4 \mathit {a0} \mathit {c2} y \left (x \right )+2 \mathit {a1} \mathit {b1} x +2 \mathit {a1} \mathit {c1} y \left (x \right )-4 \mathit {a2} \mathit {b0} x -4 \mathit {a2} \mathit {c0} y \left (x \right )-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}+\mathit {a1}}{2 \left (\mathit {a0} +\mathit {b0} x +\mathit {c0} y \left (x \right )\right )}}-\frac {{\mathrm e}^{-\left (\int -\frac {\textit {\_b}^{2} \mathit {b0} \mathit {c1} -\textit {\_b}^{2} \mathit {b1} \mathit {c0} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b1} \mathit {c2} -\mathit {b2} \mathit {c1}}{\left (\textit {\_b}^{3} \mathit {c0} +\textit {\_b}^{2} \mathit {b0} +\textit {\_b}^{2} \mathit {c1} +\textit {\_b} \mathit {b1} +\textit {\_b} \mathit {c2} +\mathit {b2} \right ) \left (\textit {\_b}^{2} \mathit {c0} +\textit {\_b} \mathit {c1} +\mathit {c2} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \mathit {a0} \mathit {c1} -\textit {\_b}^{2} \mathit {a1} \mathit {c0} +2 \textit {\_b} \mathit {a0} \mathit {c2} -2 \textit {\_b} \mathit {a2} \mathit {c0} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )}{\left (\textit {\_b}^{2} \mathit {c0} +\textit {\_b} \mathit {c1} +\mathit {c2} \right ) \left (\textit {\_b}^{3} \mathit {c0} +\textit {\_b}^{2} \mathit {b0} +\textit {\_b}^{2} \mathit {c1} +\textit {\_b} \mathit {b1} +\textit {\_b} \mathit {c2} +\mathit {b2} \right )}d \textit {\_b} +\textit {\_C1} \right ) = 0, x -{\mathrm e}^{\int _{}^{\frac {-\mathit {b1} x -\mathit {c1} y \left (x \right )-\mathit {a1} +\sqrt {-4 \mathit {b0} \mathit {b2} \,x^{2}-4 \mathit {b0} \mathit {c2} x y \left (x \right )+\mathit {b1}^{2} x^{2}+2 \mathit {b1} \mathit {c1} x y \left (x \right )-4 \mathit {b2} \mathit {c0} x y \left (x \right )-4 \mathit {c0} \mathit {c2} y \left (x \right )^{2}+\mathit {c1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {b2} x -4 \mathit {a0} \mathit {c2} y \left (x \right )+2 \mathit {a1} \mathit {b1} x +2 \mathit {a1} \mathit {c1} y \left (x \right )-4 \mathit {a2} \mathit {b0} x -4 \mathit {a2} \mathit {c0} y \left (x \right )-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}}{2 \mathit {a0} +2 \mathit {b0} x +2 \mathit {c0} y \left (x \right )}}-\frac {\textit {\_a}^{2} \mathit {b0} \mathit {c1} -\textit {\_a}^{2} \mathit {b1} \mathit {c0} +2 \textit {\_a} \mathit {b0} \mathit {c2} -2 \textit {\_a} \mathit {b2} \mathit {c0} +\mathit {b1} \mathit {c2} -\mathit {b2} \mathit {c1}}{\left (\textit {\_a}^{3} \mathit {c0} +\textit {\_a}^{2} \mathit {b0} +\textit {\_a}^{2} \mathit {c1} +\textit {\_a} \mathit {b1} +\textit {\_a} \mathit {c2} +\mathit {b2} \right ) \left (\textit {\_a}^{2} \mathit {c0} +\mathit {c1} \textit {\_a} +\mathit {c2} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-\mathit {b1} x -\mathit {c1} y \left (x \right )-\mathit {a1} +\sqrt {-4 \mathit {b0} \mathit {b2} \,x^{2}-4 \mathit {b0} \mathit {c2} x y \left (x \right )+\mathit {b1}^{2} x^{2}+2 \mathit {b1} \mathit {c1} x y \left (x \right )-4 \mathit {b2} \mathit {c0} x y \left (x \right )-4 \mathit {c0} \mathit {c2} y \left (x \right )^{2}+\mathit {c1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {b2} x -4 \mathit {a0} \mathit {c2} y \left (x \right )+2 \mathit {a1} \mathit {b1} x +2 \mathit {a1} \mathit {c1} y \left (x \right )-4 \mathit {a2} \mathit {b0} x -4 \mathit {a2} \mathit {c0} y \left (x \right )-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}}{2 \mathit {a0} +2 \mathit {b0} x +2 \mathit {c0} y \left (x \right )}}-\frac {{\mathrm e}^{-\left (\int -\frac {\textit {\_b}^{2} \mathit {b0} \mathit {c1} -\textit {\_b}^{2} \mathit {b1} \mathit {c0} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b1} \mathit {c2} -\mathit {b2} \mathit {c1}}{\left (\textit {\_b}^{3} \mathit {c0} +\textit {\_b}^{2} \mathit {b0} +\textit {\_b}^{2} \mathit {c1} +\textit {\_b} \mathit {b1} +\textit {\_b} \mathit {c2} +\mathit {b2} \right ) \left (\textit {\_b}^{2} \mathit {c0} +\textit {\_b} \mathit {c1} +\mathit {c2} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \mathit {a0} \mathit {c1} -\textit {\_b}^{2} \mathit {a1} \mathit {c0} +2 \textit {\_b} \mathit {a0} \mathit {c2} -2 \textit {\_b} \mathit {a2} \mathit {c0} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )}{\left (\textit {\_b}^{2} \mathit {c0} +\textit {\_b} \mathit {c1} +\mathit {c2} \right ) \left (\textit {\_b}^{3} \mathit {c0} +\textit {\_b}^{2} \mathit {b0} +\textit {\_b}^{2} \mathit {c1} +\textit {\_b} \mathit {b1} +\textit {\_b} \mathit {c2} +\mathit {b2} \right )}d \textit {\_b} +\textit {\_C1} \right ) = 0\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
Solve[{x == -((a2 + a1*K[2] + a0*K[2]^2 - E^RootSum[b2 + b1*#1 + c2*#1 + b0*#1^2 + c1*#1^2 + c0*#1^3 & , (c2*Log[K[2] - #1] + c1*Log[K[2] - #1]*#1 + c0*Log[K[2] - #1]*#1^2)/(b1 + c2 + 2*b0*#1 + 2*c1*#1 + 3*c0*#1^2) & ]*(c2 + K[2]*(c1 + c0*K[2]))*(C[1] + Inactive[Integrate][(-a2 - K[1]*(a1 + a0*K[1]))/(E^RootSum[b2 + b1*#1 + c2*#1 + b0*#1^2 + c1*#1^2 + c0*#1^3 & , (c2*Log[K[1] - #1] + c1*Log[K[1] - #1]*#1 + c0*Log[K[1] - #1]*#1^2)/(b1 + c2 + 2*b0*#1 + 2*c1*#1 + 3*c0*#1^2) & ]*(b2 + K[1]*(b1 + c2 + K[1]*(b0 + c1 + c0*K[1])))), {K[1], 1, K[2]}]))/(b2 + K[2]*(b1 + c2 + K[2]*(b0 + c1 + c0*K[2])))), y[x] == -((K[2]*(a2 + K[2]*(a1 + a0*K[2])) + E^RootSum[b2 + b1*#1 + c2*#1 + b0*#1^2 + c1*#1^2 + c0*#1^3 & , (c2*Log[K[2] - #1] + c1*Log[K[2] - #1]*#1 + c0*Log[K[2] - #1]*#1^2)/(b1 + c2 + 2*b0*#1 + 2*c1*#1 + 3*c0*#1^2) & ]*(b2 + K[2]*(b1 + b0*K[2]))*(C[1] + Inactive[Integrate][(-a2 - K[1]*(a1 + a0*K[1]))/(E^RootSum[b2 + b1*#1 + c2*#1 + b0*#1^2 + c1*#1^2 + c0*#1^3 & , (c2*Log[K[1] - #1] + c1*Log[K[1] - #1]*#1 + c0*Log[K[1] - #1]*#1^2)/(b1 + c2 + 2*b0*#1 + 2*c1*#1 + 3*c0*#1^2) & ]*(b2 + K[1]*(b1 + c2 + K[1]*(b0 + c1 + c0*K[1])))), {K[1], 1, K[2]}]))/(b2 + K[2]*(b1 + c2 + K[2]*(b0 + c1 + c0*K[2]))))}, {y[x], K[2]}]
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))
Maple raw output
[x-exp(Intat(-(_a^2*b0*c1-_a^2*b1*c0+2*_a*b0*c2-2*_a*b2*c0+b1*c2-b2*c1)/(_a^3*c0+_a^2*b0+_a^2*c1+_a*b1+_a*c2+b2)/(_a^2*c0+c1*_a+c2),_a = -1/2*(b1*x+c1*y(x)+(-4*b0*b2*x^2-4*b0*c2*x*y(x)+b1^2*x^2+2*b1*c1*x*y(x)-4*b2*c0*x*y(x)-4*c0*c2*y(x)^2+c1^2*y(x)^2-4*a0*b2*x-4*a0*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*a2*b0*x-4*a2*c0*y(x)-4*a0*a2+a1^2)^(1/2)+a1)/(a0+b0*x+c0*y(x))))*(Intat(-exp(-Int(-(_b^2*b0*c1-_b^2*b1*c0+2*_b*b0*c2-2*_b*b2*c0+b1*c2-b2*c1)/(_b^3*c0+_b^2*b0+_b^2*c1+_b*b1+_b*c2+b2)/(_b^2*c0+_b*c1+c2),_b))*(_b^2*a0*c1-_b^2*a1*c0+2*_b*a0*c2-2*_b*a2*c0+a1*c2-a2*c1)/(_b^2*c0+_b*c1+c2)/(_b^3*c0+_b^2*b0+_b^2*c1+_b*b1+_b*c2+b2),_b = -1/2*(b1*x+c1*y(x)+(-4*b0*b2*x^2-4*b0*c2*x*y(x)+b1^2*x^2+2*b1*c1*x*y(x)-4*b2*c0*x*y(x)-4*c0*c2*y(x)^2+c1^2*y(x)^2-4*a0*b2*x-4*a0*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*a2*b0*x-4*a2*c0*y(x)-4*a0*a2+a1^2)^(1/2)+a1)/(a0+b0*x+c0*y(x)))+_C1) = 0, x-exp(Intat(-(_a^2*b0*c1-_a^2*b1*c0+2*_a*b0*c2-2*_a*b2*c0+b1*c2-b2*c1)/(_a^3*c0+_a^2*b0+_a^2*c1+_a*b1+_a*c2+b2)/(_a^2*c0+c1*_a+c2),_a = 1/2/(a0+b0*x+c0*y(x))*(-b1*x-c1*y(x)-a1+(-4*b0*b2*x^2-4*b0*c2*x*y(x)+b1^2*x^2+2*b1*c1*x*y(x)-4*b2*c0*x*y(x)-4*c0*c2*y(x)^2+c1^2*y(x)^2-4*a0*b2*x-4*a0*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*a2*b0*x-4*a2*c0*y(x)-4*a0*a2+a1^2)^(1/2))))*(Intat(-exp(-Int(-(_b^2*b0*c1-_b^2*b1*c0+2*_b*b0*c2-2*_b*b2*c0+b1*c2-b2*c1)/(_b^3*c0+_b^2*b0+_b^2*c1+_b*b1+_b*c2+b2)/(_b^2*c0+_b*c1+c2),_b))*(_b^2*a0*c1-_b^2*a1*c0+2*_b*a0*c2-2*_b*a2*c0+a1*c2-a2*c1)/(_b^2*c0+_b*c1+c2)/(_b^3*c0+_b^2*b0+_b^2*c1+_b*b1+_b*c2+b2),_b = 1/2/(a0+b0*x+c0*y(x))*(-b1*x-c1*y(x)-a1+(-4*b0*b2*x^2-4*b0*c2*x*y(x)+b1^2*x^2+2*b1*c1*x*y(x)-4*b2*c0*x*y(x)-4*c0*c2*y(x)^2+c1^2*y(x)^2-4*a0*b2*x-4*a0*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*a2*b0*x-4*a2*c0*y(x)-4*a0*a2+a1^2)^(1/2)))+_C1) = 0]