Internal problem ID [7984]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 404.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {a \left (y^{\prime }\right )^{2}+b \,x^{2} y^{\prime }+c x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 499
dsolve(a*diff(y(x),x)^2+b*x^2*diff(y(x),x)+c*x*y(x) = 0,y(x), singsol=all)
\begin{align*} \int _{\textit {\_b}}^{x}-\frac {-b \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} a c y \relax (x )}}{-b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} a c y \relax (x )}\, \textit {\_a} -6 a y \relax (x )}d \textit {\_a} +\int _{}^{y \relax (x )}\left (\frac {2 a}{-b \,x^{3}+\sqrt {b^{2} x^{4}-4 \textit {\_f} a c x}\, x -6 a \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (-b \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\right ) \left (-\frac {2 \textit {\_a}^{2} a c}{\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}}-6 a \right )}{\left (-b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\, \textit {\_a} -6 a \textit {\_f} \right )^{2}}+\frac {2 a c \textit {\_a}}{\left (-b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\, \textit {\_a} -6 a \textit {\_f} \right ) \sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \\ \int _{\textit {\_b}}^{x}-\frac {b \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} a c y \relax (x )}}{b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} a c y \relax (x )}\, \textit {\_a} +6 a y \relax (x )}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {2 a}{b \,x^{3}+\sqrt {b^{2} x^{4}-4 \textit {\_f} a c x}\, x +6 a \textit {\_f}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (b \,\textit {\_a}^{2}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\right ) \left (-\frac {2 \textit {\_a}^{2} a c}{\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}}+6 a \right )}{\left (b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\, \textit {\_a} +6 a \textit {\_f} \right )^{2}}+\frac {2 a c \textit {\_a}}{\left (b \,\textit {\_a}^{3}+\sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}\, \textit {\_a} +6 a \textit {\_f} \right ) \sqrt {\textit {\_a}^{4} b^{2}-4 \textit {\_a} \textit {\_f} a c}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.383 (sec). Leaf size: 313
DSolve[c*x*y[x] + b*x^2*y'[x] + a*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {-6 b \tanh ^{-1}\left (\frac {b x \sqrt {b^2 x^4-4 a c x y(x)}}{b^2 x^3-4 a c y(x)}\right )+(6 b+4 c) \tanh ^{-1}\left (\frac {x^2 (3 b+2 c)}{3 \sqrt {b^2 x^4-4 a c x y(x)}}\right )+(3 b+2 c) \log \left (9 a y(x)+3 b x^3+c x^3\right )}{6 (3 b+c)}+\frac {b \log (6 b y(x)+2 c y(x))}{2 (3 b+c)}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {6 b \tanh ^{-1}\left (\frac {b x \sqrt {b^2 x^4-4 a c x y(x)}}{b^2 x^3-4 a c y(x)}\right )-2 (3 b+2 c) \tanh ^{-1}\left (\frac {x^2 (3 b+2 c)}{3 \sqrt {b^2 x^4-4 a c x y(x)}}\right )+(3 b+2 c) \log \left (9 a y(x)+3 b x^3+c x^3\right )}{6 (3 b+c)}+\frac {b \log (6 b y(x)+2 c y(x))}{2 (3 b+c)}=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}