75.23.7 problem 730

Internal problem ID [17204]
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 ODEs). Section 18.1 Integration of differential equation in series. Power series. Exercises page 171
Problem number : 730
Date solved : Tuesday, January 28, 2025 at 09:57:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \ln \left (x \right ) y^{\prime \prime }-y \sin \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} {\mathrm e} \end{align*}

With initial conditions

\begin{align*} y \left ({\mathrm e}\right )&={\mathrm e}^{-1}\\ y^{\prime }\left ({\mathrm e}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 118

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));
 

\[ y = {\mathrm e}^{-1}+\frac {1}{2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1} \left (x -{\mathrm e}\right )^{2}+\frac {1}{6} \left (\cos \left ({\mathrm e}\right ) {\mathrm e}-\sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-2} \left (x -{\mathrm e}\right )^{3}+\left (\frac {{\mathrm e}^{-3} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-3} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-3} \cos \left ({\mathrm e}\right ) {\mathrm e}}{12}\right ) \left (x -{\mathrm e}\right )^{4}-\frac {1}{120} \left (4 \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{2}-3 \sin \left ({\mathrm e}\right ) {\mathrm e}^{2}+\cos \left ({\mathrm e}\right ) {\mathrm e}^{3}-9 \cos \left ({\mathrm e}\right ) {\mathrm e}-2 \,{\mathrm e}^{3} \sin \left (2 \,{\mathrm e}\right )+14 \sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-4} \left (x -{\mathrm e}\right )^{5}+\operatorname {O}\left (\left (x -{\mathrm e}\right )^{6}\right ) \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

AsymptoticDSolveValue[{Log[x]*D[y[x],{x,2}]-Sin[x]*y[x]==0,{y[Exp[1]]==1/Exp[1],Derivative[1][y][ Exp[1] ]==0}},y[x],{x,exp(1),"6"-1}]
 

Not solved