Internal problem ID [8059]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 479.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (\mathit {b2} y+\mathit {a2} x +\mathit {c2} \right ) \left (y^{\prime }\right )^{2}+\left (\mathit {a1} x +\mathit {b1} y+\mathit {c1} \right ) y^{\prime }+\mathit {a0} x +\mathit {b0} y+\mathit {c0}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.941 (sec). Leaf size: 929
dsolve((b2*y(x)+a2*x+c2)*diff(y(x),x)^2+(a1*x+b1*y(x)+c1)*diff(y(x),x)+a0*x+b0*y(x)+c0 = 0,y(x), singsol=all)
\begin{align*} x -{\mathrm e}^{\int _{}^{-\frac {\mathit {a1} x +\mathit {b1} y \relax (x )+\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \relax (x )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \relax (x )-4 \mathit {a2} \mathit {b0} x y \relax (x )-4 \mathit {b0} \mathit {b2} y \relax (x )^{2}+\mathit {b1}^{2} y \relax (x )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \relax (x )+2 \mathit {b1} \mathit {c1} y \relax (x )-4 \mathit {b2} \mathit {c0} y \relax (x )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {b2} y \relax (x )+\mathit {a2} x +\mathit {c2} \right )}}\frac {\textit {\_a}^{2} \mathit {a1} \mathit {b2} -\textit {\_a}^{2} \mathit {a2} \mathit {b1} +2 \textit {\_a} \mathit {a0} \mathit {b2} -2 \textit {\_a} \mathit {a2} \mathit {b0} +\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0}}{\left (\textit {\_a}^{3} \mathit {b2} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a}^{2} \mathit {b1} +\textit {\_a} \mathit {a1} +\textit {\_a} \mathit {b0} +\mathit {a0} \right ) \left (\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0} \right )}d \textit {\_a}} \left (\int _{}^{-\frac {\mathit {a1} x +\mathit {b1} y \relax (x )+\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \relax (x )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \relax (x )-4 \mathit {a2} \mathit {b0} x y \relax (x )-4 \mathit {b0} \mathit {b2} y \relax (x )^{2}+\mathit {b1}^{2} y \relax (x )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \relax (x )+2 \mathit {b1} \mathit {c1} y \relax (x )-4 \mathit {b2} \mathit {c0} y \relax (x )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {c1}}{2 \left (\mathit {b2} y \relax (x )+\mathit {a2} x +\mathit {c2} \right )}}-\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} \mathit {a1} \mathit {b2} -\textit {\_b}^{2} \mathit {a2} \mathit {b1} +2 \textit {\_b} \mathit {a0} \mathit {b2} -2 \textit {\_b} \mathit {a2} \mathit {b0} +\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0}}{\left (\textit {\_b}^{3} \mathit {b2} +\textit {\_b}^{2} \mathit {a2} +\textit {\_b}^{2} \mathit {b1} +\textit {\_b} \mathit {a1} +\textit {\_b} \mathit {b0} +\mathit {a0} \right ) \left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \mathit {b1} \mathit {c2} -\textit {\_b}^{2} \mathit {b2} \mathit {c1} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0} \right )}{\left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_b}^{3} \mathit {b2} +\textit {\_b}^{2} \mathit {a2} +\textit {\_b}^{2} \mathit {b1} +\textit {\_b} \mathit {a1} +\textit {\_b} \mathit {b0} +\mathit {a0} \right )}d \textit {\_b} +c_{1}\right ) = 0 \\ x -{\mathrm e}^{\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \relax (x )-\mathit {c1} +\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \relax (x )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \relax (x )-4 \mathit {a2} \mathit {b0} x y \relax (x )-4 \mathit {b0} \mathit {b2} y \relax (x )^{2}+\mathit {b1}^{2} y \relax (x )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \relax (x )+2 \mathit {b1} \mathit {c1} y \relax (x )-4 \mathit {b2} \mathit {c0} y \relax (x )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {b2} y \relax (x )+2 \mathit {a2} x +2 \mathit {c2}}}\frac {\textit {\_a}^{2} \mathit {a1} \mathit {b2} -\textit {\_a}^{2} \mathit {a2} \mathit {b1} +2 \textit {\_a} \mathit {a0} \mathit {b2} -2 \textit {\_a} \mathit {a2} \mathit {b0} +\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0}}{\left (\textit {\_a}^{3} \mathit {b2} +\textit {\_a}^{2} \mathit {a2} +\textit {\_a}^{2} \mathit {b1} +\textit {\_a} \mathit {a1} +\textit {\_a} \mathit {b0} +\mathit {a0} \right ) \left (\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \relax (x )-\mathit {c1} +\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \relax (x )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \relax (x )-4 \mathit {a2} \mathit {b0} x y \relax (x )-4 \mathit {b0} \mathit {b2} y \relax (x )^{2}+\mathit {b1}^{2} y \relax (x )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \relax (x )+2 \mathit {b1} \mathit {c1} y \relax (x )-4 \mathit {b2} \mathit {c0} y \relax (x )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {b2} y \relax (x )+2 \mathit {a2} x +2 \mathit {c2}}}-\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} \mathit {a1} \mathit {b2} -\textit {\_b}^{2} \mathit {a2} \mathit {b1} +2 \textit {\_b} \mathit {a0} \mathit {b2} -2 \textit {\_b} \mathit {a2} \mathit {b0} +\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0}}{\left (\textit {\_b}^{3} \mathit {b2} +\textit {\_b}^{2} \mathit {a2} +\textit {\_b}^{2} \mathit {b1} +\textit {\_b} \mathit {a1} +\textit {\_b} \mathit {b0} +\mathit {a0} \right ) \left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \mathit {b1} \mathit {c2} -\textit {\_b}^{2} \mathit {b2} \mathit {c1} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0} \right )}{\left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_b}^{3} \mathit {b2} +\textit {\_b}^{2} \mathit {a2} +\textit {\_b}^{2} \mathit {b1} +\textit {\_b} \mathit {a1} +\textit {\_b} \mathit {b0} +\mathit {a0} \right )}d \textit {\_b} +c_{1}\right ) = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.893 (sec). Leaf size: 576
DSolve[c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*y'[x] + (c2 + a2*x + b2*y[x])*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=-\frac {-(K[2] (\text {b2} K[2]+\text {b1})+\text {b0}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\&,\frac {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\&\right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\&,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\&\right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )+\text {c1} K[2]+\text {c2} K[2]^2+\text {c0}}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}},y(x)=-\frac {K[2] (K[2] (\text {c2} K[2]+\text {c1})+\text {c0})+(K[2] (\text {a2} K[2]+\text {a1})+\text {a0}) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\&,\frac {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\&\right ]\right ) \left (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\&,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\&\right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}}\right \},\{y(x),K[2]\}\right ] \]