60.1.476 problem 479

Internal problem ID [10490]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 479
Date solved : Monday, January 27, 2025 at 07:55:21 PM
CAS classification : [_rational, _dAlembert]

\begin{align*} \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 }+a_{0} x +b_{0} y+c_{0}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.069 (sec). Leaf size: 875

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 {-a_{1} x -b_{1} y-\sqrt {\left (-4 b_{2} b_{0} +b_{1}^{2}\right ) y^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (2 c_{1} a_{1} -4 c_{2} a_{0} -4 a_{2} c_{0} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}-c_{1}}{2 a_{2} x +2 b_{2} y+2 c_{2}}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y-\sqrt {\left (-4 b_{2} b_{0} +b_{1}^{2}\right ) y^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (2 c_{1} a_{1} -4 c_{2} a_{0} -4 a_{2} c_{0} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}-c_{1}}{2 a_{2} x +2 b_{2} y+2 c_{2}}}\frac {\left (c_{1} \textit {\_a}^{2} b_{2} -c_{2} \textit {\_a}^{2} b_{1} -2 c_{2} \textit {\_a} b_{0} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_3 \right ) &= 0 \\ x -{\mathrm e}^{\int _{}^{\frac {-a_{1} x -b_{1} y-c_{1} +\sqrt {\left (-4 b_{2} b_{0} +b_{1}^{2}\right ) y^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (2 c_{1} a_{1} -4 c_{2} a_{0} -4 a_{2} c_{0} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 a_{2} x +2 b_{2} y+2 c_{2}}}\frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-a_{1} x -b_{1} y-c_{1} +\sqrt {\left (-4 b_{2} b_{0} +b_{1}^{2}\right ) y^{2}+\left (\left (-4 a_{0} b_{2} +2 a_{1} b_{1} -4 a_{2} b_{0} \right ) x -4 b_{2} c_{0} +2 c_{1} b_{1} -4 c_{2} b_{0} \right ) y+\left (-4 a_{0} a_{2} +a_{1}^{2}\right ) x^{2}+\left (2 c_{1} a_{1} -4 c_{2} a_{0} -4 a_{2} c_{0} \right ) x -4 c_{2} c_{0} +c_{1}^{2}}}{2 a_{2} x +2 b_{2} y+2 c_{2}}}\frac {\left (c_{1} \textit {\_a}^{2} b_{2} -c_{2} \textit {\_a}^{2} b_{1} -2 c_{2} \textit {\_a} b_{0} +2 \textit {\_a} b_{2} c_{0} -b_{0} c_{1} +c_{0} b_{1} \right ) {\mathrm e}^{-\int \frac {\left (a_{1} b_{2} -a_{2} b_{1} \right ) \textit {\_a}^{2}+\left (2 a_{0} b_{2} -2 a_{2} b_{0} \right ) \textit {\_a} +a_{0} b_{1} -b_{0} a_{1}}{\left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right ) \left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right )}d \textit {\_a}}}{\left (\textit {\_a}^{3} b_{2} +\left (a_{2} +b_{1} \right ) \textit {\_a}^{2}+\left (a_{1} +b_{0} \right ) \textit {\_a} +a_{0} \right ) \left (\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0} \right )}d \textit {\_a} +c_3 \right ) &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.886 (sec). Leaf size: 372

DSolve[c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*D[y[x],x] + (c2 + a2*x + b2*y[x])*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=-\frac {-(K[3] (\text {b2} K[3]+\text {b1})+\text {b0}) \exp \left (\int _1^{K[3]}\frac {\text {b0}+K[1] (\text {b1}+\text {b2} K[1])}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]\right ) \left (\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {b0}+K[1] (\text {b1}+\text {b2} K[1])}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]\right ) (-\text {c0}-K[2] (\text {c1}+\text {c2} K[2]))}{\text {a0}+K[2] (\text {a1}+\text {b0}+K[2] (\text {a2}+\text {b1}+\text {b2} K[2]))}dK[2]+c_1\right )+\text {c1} K[3]+\text {c2} K[3]^2+\text {c0}}{K[3] (K[3] (\text {b2} K[3]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}},y(x)=-\frac {K[3] (K[3] (\text {c2} K[3]+\text {c1})+\text {c0})+(K[3] (\text {a2} K[3]+\text {a1})+\text {a0}) \exp \left (\int _1^{K[3]}\frac {\text {b0}+K[1] (\text {b1}+\text {b2} K[1])}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]\right ) \left (\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {b0}+K[1] (\text {b1}+\text {b2} K[1])}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]\right ) (-\text {c0}-K[2] (\text {c1}+\text {c2} K[2]))}{\text {a0}+K[2] (\text {a1}+\text {b0}+K[2] (\text {a2}+\text {b1}+\text {b2} K[2]))}dK[2]+c_1\right )}{K[3] (K[3] (\text {b2} K[3]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}}\right \},\{y(x),K[3]\}\right ] \]