60.7.183 problem 1774 (book 6.183)

Internal problem ID [11772]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1774 (book 6.183)
Date solved : Tuesday, January 28, 2025 at 06:11:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y y^{\prime \prime }-x^{2} \left ({y^{\prime }}^{2}+1\right )+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.091 (sec). Leaf size: 30

dsolve(2*x^2*y(x)*diff(diff(y(x),x),x)-x^2*(diff(y(x),x)^2+1)+y(x)^2=0,y(x), singsol=all)
 

\[ y = \frac {x \left (4 c_{2}^{2} \ln \left (x \right )^{2}+4 \ln \left (x \right ) c_{1} c_{2} +c_{1}^{2}+1\right )}{4 c_{2}} \]

✓ Solution by Mathematica

Time used: 0.549 (sec). Leaf size: 49

DSolve[y[x]^2 - x^2*(1 + D[y[x],x]^2) + 2*x^2*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x \left (c_1{}^2 \log ^2(x)-2 c_2 c_1{}^2 \log (x)+4+c_2{}^2 c_1{}^2\right )}{4 c_1} \\ y(x)\to \text {Indeterminate} \\ \end{align*}