Internal problem ID [8740]
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`]]
\[ \boxed {a {y^{\prime }}^{2}+b \,x^{2} y^{\prime }+y c x=0} \]
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 479
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*} -2 a \left (\int _{}^{y \left (x \right )}\frac {1+3 \left (b \,x^{3}+6 a \textit {\_f} -\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x \right ) \left (\int _{\textit {\_b}}^{x}\frac {-\textit {\_a}^{4} b^{2}+2 \textit {\_a} \textit {\_f} a c +\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a}^{2} b}{\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \left (b \,\textit {\_a}^{3}-\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} +6 a \textit {\_f} \right )^{2}}d \textit {\_a} \right )}{b \,x^{3}+6 a \textit {\_f} -\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x}d \textit {\_f} \right )+\int _{\textit {\_b}}^{x}\frac {-b \,\textit {\_a}^{2}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y \left (x \right ) a c \right )}}{b \,\textit {\_a}^{3}+6 a y \left (x \right )-\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y \left (x \right ) a c \right )}\, \textit {\_a}}d \textit {\_a} +c_{1} &= 0 \\ -2 a \left (\int _{}^{y \left (x \right )}\frac {1+3 \left (b \,x^{3}+\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x +6 a \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a} \left (\textit {\_a}^{3} b^{2}+\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} b -2 \textit {\_f} a c \right )}{\left (b \,\textit {\_a}^{3}+\sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}\, \textit {\_a} +6 a \textit {\_f} \right )^{2} \sqrt {\textit {\_a} \left (\textit {\_a}^{3} b^{2}-4 \textit {\_f} a c \right )}}d \textit {\_a} \right )}{b \,x^{3}+\sqrt {-4 \left (-\frac {b^{2} x^{3}}{4}+\textit {\_f} a c \right ) x}\, x +6 a \textit {\_f}}d \textit {\_f} \right )-\left (\int _{\textit {\_b}}^{x}\frac {b \,\textit {\_a}^{2}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y \left (x \right ) a c \right )}}{b \,\textit {\_a}^{3}+\sqrt {-4 \textit {\_a} \left (-\frac {\textit {\_a}^{3} b^{2}}{4}+y \left (x \right ) a c \right )}\, \textit {\_a} +6 a y \left (x \right )}d \textit {\_a} \right )+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.173 (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 \text {arctanh}\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) \text {arctanh}\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 \text {arctanh}\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) \text {arctanh}\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*}