Internal problem ID [8814]
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 [_dAlembert]
\[ \boxed {\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+b_{0} y=-a_{0} x -c_{0}} \]
✓ Solution by Maple
Time used: 0.953 (sec). Leaf size: 927
dsolve((b__2*y(x)+a__2*x+c__2)*diff(y(x),x)^2+(a__1*x+b__1*y(x)+c__1)*diff(y(x),x)+a__0*x+b__0*y(x)+c__0 = 0,y(x), singsol=all)
\begin{align*} x -{\mathrm e}^{\int _{}^{-\frac {b_{1} y \left (x \right )+a_{1} x +\sqrt {-4 y \left (x \right )^{2} b_{0} b_{2} +y \left (x \right )^{2} b_{1}^{2}-4 y \left (x \right ) a_{0} b_{2} x +2 y \left (x \right ) a_{1} b_{1} x -4 y \left (x \right ) a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y \left (x \right ) c_{1} b_{1} -4 y \left (x \right ) c_{2} b_{0} -4 y \left (x \right ) b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}+c_{1}}{2 \left (b_{2} y \left (x \right )+a_{2} x +c_{2} \right )}}\frac {\textit {\_a}^{2} a_{1} b_{2} -\textit {\_a}^{2} a_{2} b_{1} +2 \textit {\_a} a_{0} b_{2} -2 \textit {\_a} a_{2} b_{0} +a_{0} b_{1} -a_{1} b_{0}}{\left (\textit {\_a}^{3} b_{2} +\textit {\_a}^{2} a_{2} +\textit {\_a}^{2} b_{1} +\textit {\_a} a_{1} +\textit {\_a} b_{0} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a}} \left (\int _{}^{-\frac {b_{1} y \left (x \right )+a_{1} x +\sqrt {-4 y \left (x \right )^{2} b_{0} b_{2} +y \left (x \right )^{2} b_{1}^{2}-4 y \left (x \right ) a_{0} b_{2} x +2 y \left (x \right ) a_{1} b_{1} x -4 y \left (x \right ) a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y \left (x \right ) c_{1} b_{1} -4 y \left (x \right ) c_{2} b_{0} -4 y \left (x \right ) b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}+c_{1}}{2 \left (b_{2} y \left (x \right )+a_{2} x +c_{2} \right )}}\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} a_{1} b_{2} -\textit {\_b}^{2} a_{2} b_{1} +2 \textit {\_b} a_{0} b_{2} -2 \textit {\_b} a_{2} b_{0} +a_{0} b_{1} -a_{1} b_{0}}{\left (\textit {\_b}^{3} b_{2} +\textit {\_b}^{2} a_{2} +\textit {\_b}^{2} b_{1} +\textit {\_b} a_{1} +\textit {\_b} b_{0} +a_{0} \right ) \left (\textit {\_b}^{2} b_{2} +\textit {\_b} b_{1} +b_{0} \right )}d \textit {\_b} \right )} \left (c_{1} \textit {\_b}^{2} b_{2} -c_{2} \textit {\_b}^{2} b_{1} -2 c_{2} \textit {\_b} b_{0} +2 \textit {\_b} b_{2} c_{0} -c_{1} b_{0} +b_{1} c_{0} \right )}{\left (\textit {\_b}^{2} b_{2} +\textit {\_b} b_{1} +b_{0} \right ) \left (\textit {\_b}^{3} b_{2} +\textit {\_b}^{2} a_{2} +\textit {\_b}^{2} b_{1} +\textit {\_b} a_{1} +\textit {\_b} b_{0} +a_{0} \right )}d \textit {\_b} +c_{3} \right ) = 0 x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {-4 y \left (x \right )^{2} b_{0} b_{2} +y \left (x \right )^{2} b_{1}^{2}-4 y \left (x \right ) a_{0} b_{2} x +2 y \left (x \right ) a_{1} b_{1} x -4 y \left (x \right ) a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y \left (x \right ) c_{1} b_{1} -4 y \left (x \right ) c_{2} b_{0} -4 y \left (x \right ) b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}}{2 b_{2} y \left (x \right )+2 a_{2} x +2 c_{2}}}\frac {\textit {\_a}^{2} a_{1} b_{2} -\textit {\_a}^{2} a_{2} b_{1} +2 \textit {\_a} a_{0} b_{2} -2 \textit {\_a} a_{2} b_{0} +a_{0} b_{1} -a_{1} b_{0}}{\left (\textit {\_a}^{3} b_{2} +\textit {\_a}^{2} a_{2} +\textit {\_a}^{2} b_{1} +\textit {\_a} a_{1} +\textit {\_a} b_{0} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y \left (x \right )-c_{1} +\sqrt {-4 y \left (x \right )^{2} b_{0} b_{2} +y \left (x \right )^{2} b_{1}^{2}-4 y \left (x \right ) a_{0} b_{2} x +2 y \left (x \right ) a_{1} b_{1} x -4 y \left (x \right ) a_{2} b_{0} x -4 a_{0} a_{2} x^{2}+a_{1}^{2} x^{2}+2 y \left (x \right ) c_{1} b_{1} -4 y \left (x \right ) c_{2} b_{0} -4 y \left (x \right ) b_{2} c_{0} +2 c_{1} a_{1} x -4 c_{2} a_{0} x -4 a_{2} c_{0} x +c_{1}^{2}-4 c_{2} c_{0}}}{2 b_{2} y \left (x \right )+2 a_{2} x +2 c_{2}}}\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} a_{1} b_{2} -\textit {\_b}^{2} a_{2} b_{1} +2 \textit {\_b} a_{0} b_{2} -2 \textit {\_b} a_{2} b_{0} +a_{0} b_{1} -a_{1} b_{0}}{\left (\textit {\_b}^{3} b_{2} +\textit {\_b}^{2} a_{2} +\textit {\_b}^{2} b_{1} +\textit {\_b} a_{1} +\textit {\_b} b_{0} +a_{0} \right ) \left (\textit {\_b}^{2} b_{2} +\textit {\_b} b_{1} +b_{0} \right )}d \textit {\_b} \right )} \left (c_{1} \textit {\_b}^{2} b_{2} -c_{2} \textit {\_b}^{2} b_{1} -2 c_{2} \textit {\_b} b_{0} +2 \textit {\_b} b_{2} c_{0} -c_{1} b_{0} +b_{1} c_{0} \right )}{\left (\textit {\_b}^{2} b_{2} +\textit {\_b} b_{1} +b_{0} \right ) \left (\textit {\_b}^{3} b_{2} +\textit {\_b}^{2} a_{2} +\textit {\_b}^{2} b_{1} +\textit {\_b} a_{1} +\textit {\_b} b_{0} +a_{0} \right )}d \textit {\_b} +c_{3} \right ) = 0 \end{align*}
✓ Solution by Mathematica
Time used: 4.811 (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 ] \]