1.477 problem 479

Internal problem ID [8057]

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]

\[ \boxed {\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right ) {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {a0} x +\operatorname {b0} y+\operatorname {c0}=0} \]

✓ Solution by Maple

Time used: 0.765 (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 {\operatorname {a1} x +\operatorname {b1} y \left (x \right )+\sqrt {-4 \operatorname {a0} \operatorname {a2} \,x^{2}-4 \operatorname {a0} \operatorname {b2} x y \left (x \right )+\operatorname {a1}^{2} x^{2}+2 \operatorname {a1} \operatorname {b1} x y \left (x \right )-4 \operatorname {a2} \operatorname {b0} x y \left (x \right )-4 \operatorname {b0} \operatorname {b2} y \left (x \right )^{2}+\operatorname {b1}^{2} y \left (x \right )^{2}-4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {b0} \operatorname {c2} y \left (x \right )+2 \operatorname {b1} \operatorname {c1} y \left (x \right )-4 \operatorname {b2} \operatorname {c0} y \left (x \right )-4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}+\operatorname {c1}}{2 \left (\operatorname {b2} y \left (x \right )+\operatorname {a2} x +\operatorname {c2} \right )}}\frac {\textit {\_a}^{2} \operatorname {a1} \operatorname {b2} -\textit {\_a}^{2} \operatorname {a2} \operatorname {b1} +2 \textit {\_a} \operatorname {a0} \operatorname {b2} -2 \textit {\_a} \operatorname {a2} \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\textit {\_a}^{3} \operatorname {b2} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a}^{2} \operatorname {b1} +\textit {\_a} \operatorname {a1} +\textit {\_a} \operatorname {b0} +\operatorname {a0} \right ) \left (\textit {\_a}^{2} \operatorname {b2} +\textit {\_a} \operatorname {b1} +\operatorname {b0} \right )}d \textit {\_a}} \left (\int _{}^{-\frac {\operatorname {a1} x +\operatorname {b1} y \left (x \right )+\sqrt {-4 \operatorname {a0} \operatorname {a2} \,x^{2}-4 \operatorname {a0} \operatorname {b2} x y \left (x \right )+\operatorname {a1}^{2} x^{2}+2 \operatorname {a1} \operatorname {b1} x y \left (x \right )-4 \operatorname {a2} \operatorname {b0} x y \left (x \right )-4 \operatorname {b0} \operatorname {b2} y \left (x \right )^{2}+\operatorname {b1}^{2} y \left (x \right )^{2}-4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {b0} \operatorname {c2} y \left (x \right )+2 \operatorname {b1} \operatorname {c1} y \left (x \right )-4 \operatorname {b2} \operatorname {c0} y \left (x \right )-4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}+\operatorname {c1}}{2 \left (\operatorname {b2} y \left (x \right )+\operatorname {a2} x +\operatorname {c2} \right )}}-\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} \operatorname {a1} \operatorname {b2} -\textit {\_b}^{2} \operatorname {a2} \operatorname {b1} +2 \textit {\_b} \operatorname {a0} \operatorname {b2} -2 \textit {\_b} \operatorname {a2} \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\textit {\_b}^{3} \operatorname {b2} +\textit {\_b}^{2} \operatorname {a2} +\textit {\_b}^{2} \operatorname {b1} +\textit {\_b} \operatorname {a1} +\textit {\_b} \operatorname {b0} +\operatorname {a0} \right ) \left (\textit {\_b}^{2} \operatorname {b2} +\textit {\_b} \operatorname {b1} +\operatorname {b0} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \operatorname {b1} \operatorname {c2} -\textit {\_b}^{2} \operatorname {b2} \operatorname {c1} +2 \textit {\_b} \operatorname {b0} \operatorname {c2} -2 \textit {\_b} \operatorname {b2} \operatorname {c0} +\operatorname {b0} \operatorname {c1} -\operatorname {b1} \operatorname {c0} \right )}{\left (\textit {\_b}^{2} \operatorname {b2} +\textit {\_b} \operatorname {b1} +\operatorname {b0} \right ) \left (\textit {\_b}^{3} \operatorname {b2} +\textit {\_b}^{2} \operatorname {a2} +\textit {\_b}^{2} \operatorname {b1} +\textit {\_b} \operatorname {a1} +\textit {\_b} \operatorname {b0} +\operatorname {a0} \right )}d \textit {\_b} +c_{1} \right ) = 0 \\ x -{\mathrm e}^{\int _{}^{\frac {-\operatorname {a1} x -\operatorname {b1} y \left (x \right )-\operatorname {c1} +\sqrt {-4 \operatorname {a0} \operatorname {a2} \,x^{2}-4 \operatorname {a0} \operatorname {b2} x y \left (x \right )+\operatorname {a1}^{2} x^{2}+2 \operatorname {a1} \operatorname {b1} x y \left (x \right )-4 \operatorname {a2} \operatorname {b0} x y \left (x \right )-4 \operatorname {b0} \operatorname {b2} y \left (x \right )^{2}+\operatorname {b1}^{2} y \left (x \right )^{2}-4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {b0} \operatorname {c2} y \left (x \right )+2 \operatorname {b1} \operatorname {c1} y \left (x \right )-4 \operatorname {b2} \operatorname {c0} y \left (x \right )-4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}}{2 \operatorname {b2} y \left (x \right )+2 \operatorname {a2} x +2 \operatorname {c2}}}\frac {\textit {\_a}^{2} \operatorname {a1} \operatorname {b2} -\textit {\_a}^{2} \operatorname {a2} \operatorname {b1} +2 \textit {\_a} \operatorname {a0} \operatorname {b2} -2 \textit {\_a} \operatorname {a2} \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\textit {\_a}^{3} \operatorname {b2} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a}^{2} \operatorname {b1} +\textit {\_a} \operatorname {a1} +\textit {\_a} \operatorname {b0} +\operatorname {a0} \right ) \left (\textit {\_a}^{2} \operatorname {b2} +\textit {\_a} \operatorname {b1} +\operatorname {b0} \right )}d \textit {\_a}} \left (\int _{}^{\frac {-\operatorname {a1} x -\operatorname {b1} y \left (x \right )-\operatorname {c1} +\sqrt {-4 \operatorname {a0} \operatorname {a2} \,x^{2}-4 \operatorname {a0} \operatorname {b2} x y \left (x \right )+\operatorname {a1}^{2} x^{2}+2 \operatorname {a1} \operatorname {b1} x y \left (x \right )-4 \operatorname {a2} \operatorname {b0} x y \left (x \right )-4 \operatorname {b0} \operatorname {b2} y \left (x \right )^{2}+\operatorname {b1}^{2} y \left (x \right )^{2}-4 \operatorname {a0} \operatorname {c2} x +2 \operatorname {a1} \operatorname {c1} x -4 \operatorname {a2} \operatorname {c0} x -4 \operatorname {b0} \operatorname {c2} y \left (x \right )+2 \operatorname {b1} \operatorname {c1} y \left (x \right )-4 \operatorname {b2} \operatorname {c0} y \left (x \right )-4 \operatorname {c0} \operatorname {c2} +\operatorname {c1}^{2}}}{2 \operatorname {b2} y \left (x \right )+2 \operatorname {a2} x +2 \operatorname {c2}}}-\frac {{\mathrm e}^{-\left (\int \frac {\textit {\_b}^{2} \operatorname {a1} \operatorname {b2} -\textit {\_b}^{2} \operatorname {a2} \operatorname {b1} +2 \textit {\_b} \operatorname {a0} \operatorname {b2} -2 \textit {\_b} \operatorname {a2} \operatorname {b0} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0}}{\left (\textit {\_b}^{3} \operatorname {b2} +\textit {\_b}^{2} \operatorname {a2} +\textit {\_b}^{2} \operatorname {b1} +\textit {\_b} \operatorname {a1} +\textit {\_b} \operatorname {b0} +\operatorname {a0} \right ) \left (\textit {\_b}^{2} \operatorname {b2} +\textit {\_b} \operatorname {b1} +\operatorname {b0} \right )}d \textit {\_b} \right )} \left (\textit {\_b}^{2} \operatorname {b1} \operatorname {c2} -\textit {\_b}^{2} \operatorname {b2} \operatorname {c1} +2 \textit {\_b} \operatorname {b0} \operatorname {c2} -2 \textit {\_b} \operatorname {b2} \operatorname {c0} +\operatorname {b0} \operatorname {c1} -\operatorname {b1} \operatorname {c0} \right )}{\left (\textit {\_b}^{2} \operatorname {b2} +\textit {\_b} \operatorname {b1} +\operatorname {b0} \right ) \left (\textit {\_b}^{3} \operatorname {b2} +\textit {\_b}^{2} \operatorname {a2} +\textit {\_b}^{2} \operatorname {b1} +\textit {\_b} \operatorname {a1} +\textit {\_b} \operatorname {b0} +\operatorname {a0} \right )}d \textit {\_b} +c_{1} \right ) = 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.141 (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 ] \]