69.23.7 problem 730
Internal
problem
ID
[18491]
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
:
Thursday, October 02, 2025 at 03:14:15 PM
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*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 116
Order:=6;
ode:=ln(x)*diff(diff(y(x),x),x)-y(x)*sin(x) = 0;
ic:=[y(exp(1)) = 1/exp(1), D(y)(exp(1)) = 0];
dsolve([ode,op(ic)],y(x),type='series',x=exp(1));
\[
y = {\mathrm e}^{-1}+\frac {1}{2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1} \left (x -{\mathrm e}\right )^{2}+\frac {1}{6} \left (\cos \left ({\mathrm e}\right ) {\mathrm e}-\sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-2} \left (x -{\mathrm e}\right )^{3}+\left (\frac {\sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}+\frac {\left (-{\mathrm e}^{-1}+3 \,{\mathrm e}^{-3}\right ) \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-2} \cos \left ({\mathrm e}\right )}{12}\right ) \left (x -{\mathrm e}\right )^{4}-\frac {1}{120} \left (\left (-2 \sin \left (2 \,{\mathrm e}\right )+\cos \left ({\mathrm e}\right )\right ) {\mathrm e}^{3}+\left (4 \sin \left ({\mathrm e}\right )^{2}-3 \sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{2}-9 \cos \left ({\mathrm e}\right ) {\mathrm e}+14 \sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-4} \left (x -{\mathrm e}\right )^{5}+\operatorname {O}\left (\left (x -{\mathrm e}\right )^{6}\right )
\]
✓ Mathematica. Time used: 0.013 (sec). Leaf size: 160
ode=Log[x]*D[y[x],{x,2}]-Sin[x]*y[x]==0;
ic={y[Exp[1]]==1/Exp[1],Derivative[1][y][ Exp[1] ]==0};
AsymptoticDSolveValue[{ode,ic},y[x],{x,Exp[1],5}]
\[
y(x)\to \frac {(x-e)^2 \sin (e)}{2 e}+\frac {1}{12} (x-e)^4 \left (\frac {\sin ^2(e)}{2 e}-\frac {\frac {\sin (e)}{2}-\frac {3 \sin (e)}{2 e^2}+\frac {\cos (e)}{e}}{e}\right )+\frac {1}{20} (x-e)^5 \left (-\frac {2 \sin (e) \left (\frac {\sin (e)}{e}-\cos (e)\right )}{3 e}-\frac {14 \sin (e)-3 e^2 \sin (e)-9 e \cos (e)+e^3 \cos (e)}{6 e^4}\right )-\frac {(x-e)^3 \left (\frac {\sin (e)}{e}-\cos (e)\right )}{6 e}+\frac {1}{e}
\]
✓ Sympy. Time used: 1.509 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x)*sin(x) + log(x)*Derivative(y(x), (x, 2)),0)
ics = {y(E): exp(-1), Subs(Derivative(y(x), x), x, E): 0}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=E,n=6)
\[
y{\left (x \right )} = C_{2} \left (\frac {\left (x - e\right )^{4} \sin ^{2}{\left (x + e \right )}}{24 \log {\left (x + e \right )}^{2}} + \frac {\left (x - e\right )^{2} \sin {\left (x + e \right )}}{2 \log {\left (x + e \right )}} + 1\right ) + C_{1} \left (x + \frac {\left (x - e\right )^{3} \sin {\left (x + e \right )}}{6 \log {\left (x + e \right )}} - e\right ) + O\left (x^{6}\right )
\]