73.24.3 problem 34.5 (c)
Internal
problem
ID
[15604]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
34.
Power
series
solutions
II:
Generalization
and
theory.
Additional
Exercises.
page
678
Problem
number
:
34.5
(c)
Date
solved
:
Thursday, March 13, 2025 at 06:12:17 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-{\mathrm e}^{x} y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 2 \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 289
Order:=6;
ode:=sin(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-exp(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=2);
\[
y = \left (1+\frac {\csc \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{2}-\frac {\csc \left (2\right )^{2} {\mathrm e}^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) \left (x -2\right )^{3}}{6}+\frac {\left (\left (\cos \left (2\right )-\sin \left (2\right )-\frac {\sin \left (4\right )}{12}+\frac {3}{2}\right ) {\mathrm e}^{2}+\frac {{\mathrm e}^{4} \sin \left (2\right )}{12}\right ) \csc \left (2\right )^{3} \left (x -2\right )^{4}}{2}+\frac {\csc \left (2\right )^{4} \left (\left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right ) {\mathrm e}^{2}+8 \left (-\frac {\sin \left (4\right )}{2}+\sin \left (2\right )^{2}-2 \sin \left (2\right )\right ) {\mathrm e}^{4}\right ) \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\csc \left (2\right )^{2} \left (-\sin \left (2\right ) {\mathrm e}^{2}-16-4 \cos \left (2\right )+4 \sin \left (2\right )\right ) \left (x -2\right )^{3}}{6}+\frac {\left (\left (-\frac {\cos \left (4\right )}{12}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}+\frac {1}{12}\right ) {\mathrm e}^{2}-4 \cos \left (2\right )-\frac {\cos \left (4\right )}{12}+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {71}{12}\right ) \csc \left (2\right )^{3} \left (x -2\right )^{4}}{2}+\frac {\csc \left (2\right )^{4} \left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \,{\mathrm e}^{4} \sin \left (2\right )^{2}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \left (x -2\right )^{5}}{240}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 953
ode=Sin[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Exp[x]*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 2.859 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x) - y(x)*exp(x) + sin(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sin {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sin ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sin ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} e^{x + 2}}{6 \sin {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sin {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sin ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sin {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{4} e^{2 x + 4}}{24 \sin ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} e^{x + 2}}{2 \sin {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]