7.176 problem 1767 (book 6.176)

Internal problem ID [9341]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1767 (book 6.176).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\[ \boxed {y^{\prime \prime } x y-4 x {y^{\prime }}^{2}+4 y^{\prime } y=0} \]

Solution by Maple

Time used: 0.094 (sec). Leaf size: 88

dsolve(x*y(x)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)^2+4*y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = \frac {x}{\left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) = \left (-\frac {1}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}}\right ) x \\ y \left (x \right ) = \left (-\frac {1}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{\frac {1}{3}}}\right ) x \\ \end{align*}

Solution by Mathematica

Time used: 0.118 (sec). Leaf size: 21

DSolve[4*y[x]*y'[x] - 4*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {c_2 x}{\sqrt [3]{1+c_1 x^3}} \\ \end{align*}