67.24.3 problem 34.5 (c)
Internal
problem
ID
[16970]
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, October 02, 2025 at 01:40:47 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sin \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 2 \end{align*}
✓ Maple. Time used: 0.026 (sec). Leaf size: 292
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);
\[
y = \left (1+\frac {{\mathrm e}^{2} \csc \left (2\right ) \left (x -2\right )^{2}}{2}-\frac {\csc \left (2\right )^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) {\mathrm e}^{2} \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (\cos \left (2\right )-\sin \left (2\right )-\frac {\sin \left (4\right )}{12}+\frac {3}{2}\right ) {\mathrm e}^{2}+\frac {\sin \left (2\right ) {\mathrm e}^{4}}{12}\right ) \left (x -2\right )^{4}}{2}+\frac {\left ({\mathrm e}^{2} \left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right )+8 \,{\mathrm e}^{4} \left (-\frac {\sin \left (4\right )}{2}+\sin \left (2\right )^{2}-2 \sin \left (2\right )\right )\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\csc \left (2\right )^{2} \left (-{\mathrm e}^{2} \sin \left (2\right )-4 \cos \left (2\right )+4 \sin \left (2\right )-16\right ) \left (x -2\right )^{3}}{6}+\frac {\csc \left (2\right )^{3} \left (\left (\frac {\sin \left (2\right )^{2}}{6}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}\right ) {\mathrm e}^{2}-\frac {\cos \left (2\right )^{2}}{6}-4 \cos \left (2\right )+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {35}{6}\right ) \left (x -2\right )^{4}}{2}+\frac {\left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \,{\mathrm e}^{4} \sin \left (2\right )^{2}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \csc \left (2\right )^{4} \left (x -2\right )^{5}}{240}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.039 (sec). Leaf size: 649
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}]
\[
y(x)\to c_2 \left (x+\frac {32}{15} (x-2)^5 \csc ^4(2)+\frac {2}{5} e^2 (x-2)^5 \csc ^3(2)-\frac {8}{3} (x-2)^4 \csc ^3(2)+\frac {1}{120} e^4 (x-2)^5 \csc ^2(2)-\frac {1}{3} e^2 (x-2)^4 \csc ^2(2)+\frac {8}{3} (x-2)^3 \csc ^2(2)+\frac {1}{6} e^2 (x-2)^3 \csc (2)-2 (x-2)^2 \csc (2)+\frac {1}{15} (x-2)^5 \left (3+4 \cot ^2(2)-4 \cot (2)\right ) \csc ^2(2)-\frac {1}{5} (x-2)^5 \left (-3-4 \cot ^2(2)+4 \cot (2)\right ) \csc ^2(2)-\frac {1}{12} (x-2)^4 \left (3+4 \cot ^2(2)-4 \cot (2)\right ) \csc (2)+\frac {16}{5} (x-2)^5 (\cot (2)-1) \csc ^3(2)+\frac {2}{5} (x-2)^5 (\cot (2)-1)^2 \csc ^2(2)+\frac {3}{10} e^2 (x-2)^5 (\cot (2)-1) \csc ^2(2)-2 (x-2)^4 (\cot (2)-1) \csc ^2(2)-\frac {1}{12} e^2 (x-2)^4 (\cot (2)-1) \csc (2)+\frac {2}{3} (x-2)^3 (\cot (2)-1) \csc (2)-\frac {1}{40} e^2 (x-2)^5 (\sin (4)-2) \csc ^3(2)+\frac {1}{240} (x-2)^5 \csc ^3(2) (-30+49 \cot (2)-\cos (6) \csc (2)-6 \sin (6) \csc (2))-2\right )+c_1 \left (-\frac {8}{15} e^2 (x-2)^5 \csc ^4(2)-\frac {1}{15} e^4 (x-2)^5 \csc ^3(2)+\frac {2}{3} e^2 (x-2)^4 \csc ^3(2)+\frac {1}{24} e^4 (x-2)^4 \csc ^2(2)-\frac {2}{3} e^2 (x-2)^3 \csc ^2(2)+\frac {1}{2} e^2 (x-2)^2 \csc (2)+\frac {1}{20} e^2 (x-2)^5 \left (-3-4 \cot ^2(2)+4 \cot (2)\right ) \csc ^2(2)-\frac {4}{5} e^2 (x-2)^5 (\cot (2)-1) \csc ^3(2)-\frac {1}{10} e^2 (x-2)^5 (\cot (2)-1)^2 \csc ^2(2)-\frac {1}{30} e^4 (x-2)^5 (\cot (2)-1) \csc ^2(2)+\frac {1}{2} e^2 (x-2)^4 (\cot (2)-1) \csc ^2(2)-\frac {1}{6} e^2 (x-2)^3 (\cot (2)-1) \csc (2)+\frac {1}{30} e^2 (x-2)^5 (\sin (4)-2) \csc ^4(2)-\frac {1}{24} e^2 (x-2)^4 (\sin (4)-2) \csc ^3(2)-\frac {1}{240} e^2 (x-2)^5 \csc ^3(2) (-9+13 \cot (2)-\cos (6) \csc (2)-\sin (6) \csc (2))+1\right )
\]
✓ Sympy. Time used: 1.514 (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)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sin {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sin ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sin ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} e^{x + 2}}{6 \sin {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sin {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sin ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sin {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} e^{x + 2}}{3 \sin ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{4} e^{2 x + 4}}{24 \sin ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} e^{x + 2}}{3 \sin ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} e^{x + 2}}{2 \sin {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]