Internal problem ID [15474]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 18.1 Integration of differential equation in
series. Power series. Exercises page 171
Problem number: 730.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\ln \left (x \right ) y^{\prime \prime }-y \sin \left (x \right )=0} \] With initial conditions \begin {align*} [y \left ({\mathrm e}\right ) = {\mathrm e}^{-1}, y^{\prime }\left ({\mathrm e}\right ) = 0] \end {align*}
With the expansion point for the power series method at \(x = {\mathrm e}\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 142
Order:=6; dsolve([ln(x)*diff(y(x),x$2)-y(x)*sin(x)=0,y(exp(1)) = 1/exp(1), D(y)(exp(1)) = 0],y(x),type='series',x=exp(1));
\[ y \left (x \right ) = {\mathrm e}^{-1}+\frac {1}{2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1} \left (x -{\mathrm e}\right )^{2}+\frac {1}{6} \left (\cos \left ({\mathrm e}\right ) {\mathrm e}-\sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-2} \left (x -{\mathrm e}\right )^{3}+\left (\frac {{\mathrm e}^{-3} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-3} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-3} \cos \left ({\mathrm e}\right ) {\mathrm e}}{12}\right ) \left (x -{\mathrm e}\right )^{4}+\left (-\frac {{\mathrm e}^{-4} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {\left (4 \cos \left ({\mathrm e}\right ) {\mathrm e}^{3}+3 \,{\mathrm e}^{2}-14\right ) {\mathrm e}^{-4} \sin \left ({\mathrm e}\right )}{120}+\frac {3 \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4} \left ({\mathrm e}-\frac {{\mathrm e}^{3}}{9}\right )}{40}\right ) \left (x -{\mathrm e}\right )^{5}+\operatorname {O}\left (\left (x -{\mathrm e}\right )^{6}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
AsymptoticDSolveValue[{Log[x]*y''[x]-Sin[x]*y[x]==0,{y[Exp[1]]==1/Exp[1],y'[Exp[1]]==0}},y[x],{x,exp(1),5}]
Not solved