56.1.87 problem 86

Internal problem ID [8799]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 86
Date solved : Monday, January 27, 2025 at 05:01:37 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 56

dsolve(diff(y(x),x$2)=1/y(x)-x/y(x)^2*diff(y(x),x),y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left (\left (4 \,{\mathrm e}^{\textit {\_Z}} {\cosh \left (\frac {\sqrt {c_{1}^{2}+4}\, \left (2 c_{2} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 c_{1}}\right )}^{2}+c_{1}^{2}+4\right ) x^{2}\right )}-1+\textit {\_Z} c_{1} \right ) x \]

✓ Solution by Mathematica

Time used: 0.200 (sec). Leaf size: 77

DSolve[D[y[x],{x,2}]==1/y[x]-x/y[x]^2*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {1}{2} \log \left (-\frac {y(x)^2}{x^2}-\frac {c_1 y(x)}{x}+1\right )-\frac {c_1 \arctan \left (\frac {\frac {2 y(x)}{x}+c_1}{\sqrt {-4-c_1{}^2}}\right )}{\sqrt {-4-c_1{}^2}}=-\log (x)+c_2,y(x)\right ] \]