77.1.94 problem 121 (page 179)

Internal problem ID [17984]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 121 (page 179)
Date solved : Tuesday, January 28, 2025 at 11:19:25 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (y x +1\right ) y^{\prime \prime }+x^{2} {y^{\prime }}^{2}+\left (4 y x +2\right ) y^{\prime }+y^{2}+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.104 (sec). Leaf size: 53

dsolve(x*(x*y(x)+1)*diff(y(x),x$2)+x^2*diff(y(x),x)^2+(4*x*y(x)+2)*diff(y(x),x)+y(x)^2+1=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {-1-\sqrt {-2 c_{1} x -x^{2}+2 c_{2} +1}}{x} \\ y &= \frac {-1+\sqrt {-2 c_{1} x -x^{2}+2 c_{2} +1}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.523 (sec). Leaf size: 84

DSolve[x*(x*y[x]+1)*D[y[x],{x,2}]+x^2*D[y[x],x]^2+(4*x*y[x]+2)*D[y[x],x]+y[x]^2+1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1+\sqrt {\frac {1}{x^2}} \sqrt {-x^2 \left (x^2-c_2 x-1-2 c_1\right )}}{x} \\ y(x)\to \frac {-1+\sqrt {\frac {1}{x^2}} \sqrt {-x^2 \left (x^2-c_2 x-1-2 c_1\right )}}{x} \\ \end{align*}