Internal problem ID [8047]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 467.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}-4 x y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.333 (sec). Leaf size: 91
dsolve(y(x)*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2}+4}+2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} \right )+c_{1}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 45.801 (sec). Leaf size: 182
DSolve[y[x] - 4*x*y'[x] + y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {\frac {8\ 2^{2/3} x^4+\sqrt [3]{2} \left (32 x^6-40 c_1{}^3 x^3+\sqrt {\left (c_1{}^4-16 c_1 x^3\right ){}^3}-c_1{}^6\right ){}^{2/3}+4 x^2 \sqrt [3]{32 x^6-40 c_1{}^3 x^3+\sqrt {\left (c_1{}^4-16 c_1 x^3\right ){}^3}-c_1{}^6}+4\ 2^{2/3} c_1{}^3 x}{\sqrt [3]{32 x^6-40 c_1{}^3 x^3+\sqrt {\left (c_1{}^4-16 c_1 x^3\right ){}^3}-c_1{}^6}}} \\ y(x)\to 0 \\ \end{align*}