Internal problem ID [8081]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 501.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {\left (a y^{2}+b x +c \right ) \left (y^{\prime }\right )^{2}-y^{\prime } y b +d y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 2.406 (sec). Leaf size: 291
dsolve((a*y(x)^2+b*x+c)*diff(y(x),x)^2-b*y(x)*diff(y(x),x)+d*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \left [x \left (\textit {\_T} \right ) = -\frac {\ln \relax (2)^{2} \sqrt {\textit {\_T}^{2} a +d}\, b^{2}+2 \ln \relax (2) \ln \left (\frac {\sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}+d}{\textit {\_T}}\right ) \sqrt {\textit {\_T}^{2} a +d}\, b^{2}+4 \ln \relax (2) \sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}\, c_{1} b +\ln \left (\frac {\sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}+d}{\textit {\_T}}\right )^{2} \sqrt {\textit {\_T}^{2} a +d}\, b^{2}+4 \ln \left (\frac {\sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}+d}{\textit {\_T}}\right ) \sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}\, c_{1} b +4 d \sqrt {\textit {\_T}^{2} a +d}\, c_{1}^{2}-2 \ln \relax (2) \sqrt {d}\, b^{2}-2 \ln \left (\frac {\sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}+d}{\textit {\_T}}\right ) \sqrt {d}\, b^{2}+4 c d \sqrt {\textit {\_T}^{2} a +d}-4 d c_{1} b}{4 b \sqrt {\textit {\_T}^{2} a +d}\, d}, y \left (\textit {\_T} \right ) = \frac {\left (b \ln \relax (2)+b \ln \left (\frac {\sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}+d}{\textit {\_T}}\right )+2 c_{1} \sqrt {d}\right ) \textit {\_T}}{2 \sqrt {d}\, \sqrt {\textit {\_T}^{2} a +d}}\right ] \\ \end{align*}
✓ Solution by Mathematica
Time used: 99.793 (sec). Leaf size: 980
DSolve[d*y[x]^2 - b*y[x]*y'[x] + (c + b*x + a*y[x]^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\left \{y(x)=\frac {b K[1]-\sqrt {-K[1]^2 \left (4 a b x K[1]^2+4 a c K[1]^2-b^2+4 b d x+4 c d\right )}}{2 \left (a K[1]^2+d\right )},x=\frac {-2 b^2 c_1 d^{5/2} \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 a b^2 c_1 d^{3/2} K[1]^2 \log (K[1])-2 a b^2 c_1 d^{3/2} K[1]^2 \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 c_1{}^2 d^3 K[1]^2+2 b^2 c_1 d^2 \sqrt {a K[1]^2+d}-b^2 d \log ^2\left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 K[1]^2 \log ^2\left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 b^2 d \log (K[1]) \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-2 b^2 \sqrt {d} \log (K[1]) \sqrt {a K[1]^2+d}+2 b^2 \sqrt {d} \sqrt {a K[1]^2+d} \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 a b^2 K[1]^2 \log (K[1]) \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 K[1]^2 \log ^2(K[1])-4 a c d K[1]^2+2 b^2 c_1 d^{5/2} \log (K[1])-b^2 d \log ^2(K[1])-b^2 c_1{}^2 d^4-4 c d^2}{4 b d \left (a K[1]^2+d\right )}\right \},\{y(x),K[1]\}\right ] \\ \text {Solve}\left [\left \{y(x)=\frac {\sqrt {-K[2]^2 \left (4 a b x K[2]^2+4 a c K[2]^2-b^2+4 b d x+4 c d\right )}+b K[2]}{2 \left (a K[2]^2+d\right )},x=\frac {-2 b^2 c_1 d^{5/2} \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 a b^2 c_1 d^{3/2} K[2]^2 \log (K[2])-2 a b^2 c_1 d^{3/2} K[2]^2 \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 c_1{}^2 d^3 K[2]^2+2 b^2 c_1 d^2 \sqrt {a K[2]^2+d}-b^2 d \log ^2\left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 K[2]^2 \log ^2\left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 b^2 d \log (K[2]) \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-2 b^2 \sqrt {d} \log (K[2]) \sqrt {a K[2]^2+d}+2 b^2 \sqrt {d} \sqrt {a K[2]^2+d} \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 a b^2 K[2]^2 \log (K[2]) \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 K[2]^2 \log ^2(K[2])-4 a c d K[2]^2+2 b^2 c_1 d^{5/2} \log (K[2])-b^2 d \log ^2(K[2])-b^2 c_1{}^2 d^4-4 c d^2}{4 b d \left (a K[2]^2+d\right )}\right \},\{y(x),K[2]\}\right ] \\ y(x)\to 0 \\ \end{align*}