Internal problem ID [6378]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 86.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {1}{y}+\frac {x y^{\prime }}{y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 88
dsolve(diff(y(x),x$2)=1/y(x)-x/y(x)^2*diff(y(x),x),y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (\textit {\_Z}^{2}-{\mathrm e}^{\RootOf \left (x^{2} \left (\left (\tanh ^{2}\left (\frac {\sqrt {c_{1}^{2}+4}\, \left (2 c_{2}+\textit {\_Z} +2 \ln \relax (x )\right )}{2 c_{1}}\right )\right ) c_{1}^{2}+4 \left (\tanh ^{2}\left (\frac {\sqrt {c_{1}^{2}+4}\, \left (2 c_{2}+\textit {\_Z} +2 \ln \relax (x )\right )}{2 c_{1}}\right )\right )-c_{1}^{2}-4 \,{\mathrm e}^{\textit {\_Z}}-4\right )\right )}-1+c_{1} \textit {\_Z} \right ) x \]
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 77
DSolve[y''[x]==1/y[x]-x/y[x]^2*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \log \left (-\frac {y(x)^2}{x^2}-\frac {c_1 y(x)}{x}+1\right )-\frac {c_1 \text {ArcTan}\left (\frac {\frac {2 y(x)}{x}+c_1}{\sqrt {-4-c_1{}^2}}\right )}{\sqrt {-4-c_1{}^2}}=-\log (x)+c_2,y(x)\right ] \]