73.24.3 problem 34.5 (c)
Internal
problem
ID
[15683]
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
:
Tuesday, January 28, 2025 at 08:04:08 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*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 289
Order:=6;
dsolve(sin(x)*diff(y(x),x$2)+x^2*diff(y(x),x)-exp(x)*y(x)=0,y(x),type='series',x=2);
\[
y = \left (1+\frac {\csc \left (2\right ) {\mathrm e}^{2} \left (x -2\right )^{2}}{2}-\frac {\csc \left (2\right )^{2} {\mathrm e}^{2} \left (4+\cos \left (2\right )-\sin \left (2\right )\right ) \left (x -2\right )^{3}}{6}+\frac {\left (\left (\cos \left (2\right )-\sin \left (2\right )-\frac {\sin \left (4\right )}{12}+\frac {3}{2}\right ) {\mathrm e}^{2}+\frac {{\mathrm e}^{4} \sin \left (2\right )}{12}\right ) \csc \left (2\right )^{3} \left (x -2\right )^{4}}{2}+\frac {\csc \left (2\right )^{4} \left (\left (-210+56 \sin \left (4\right )+\sin \left (6\right )+201 \sin \left (2\right )+\cos \left (6\right )-205 \cos \left (2\right )-6 \cos \left (4\right )\right ) {\mathrm e}^{2}+8 \left (-\frac {\sin \left (4\right )}{2}+\sin \left (2\right )^{2}-2 \sin \left (2\right )\right ) {\mathrm e}^{4}\right ) \left (x -2\right )^{5}}{240}\right ) y \left (2\right )+\left (x -2-2 \csc \left (2\right ) \left (x -2\right )^{2}-\frac {\csc \left (2\right )^{2} \left (-\sin \left (2\right ) {\mathrm e}^{2}-16-4 \cos \left (2\right )+4 \sin \left (2\right )\right ) \left (x -2\right )^{3}}{6}+\frac {\left (\left (-\frac {\cos \left (4\right )}{12}-\frac {2 \sin \left (2\right )}{3}-\frac {\sin \left (4\right )}{12}+\frac {1}{12}\right ) {\mathrm e}^{2}-4 \cos \left (2\right )-\frac {\cos \left (4\right )}{12}+4 \sin \left (2\right )+\frac {\sin \left (4\right )}{3}-\frac {71}{12}\right ) \csc \left (2\right )^{3} \left (x -2\right )^{4}}{2}+\frac {\csc \left (2\right )^{4} \left (\left (\left (-12 \cos \left (2\right )-72\right ) \sin \left (2\right )^{2}+108 \sin \left (2\right )+36 \sin \left (4\right )\right ) {\mathrm e}^{2}+2 \,{\mathrm e}^{4} \sin \left (2\right )^{2}-6 \sin \left (6\right )+817 \cos \left (2\right )+32 \cos \left (4\right )-\cos \left (6\right )-798 \sin \left (2\right )-224 \sin \left (4\right )+832\right ) \left (x -2\right )^{5}}{240}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 953
AsymptoticDSolveValue[Sin[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Exp[x]*y[x]==0,y[x],{x,2,"6"-1}]
\[
y(x)\to c_2 \left (-\frac {1}{60} \left (6 \csc (2)-13 \cot (2) \csc (2)+12 \cot ^2(2) \csc (2)-12 \cot ^3(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {4}{15} \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (4 \csc (2)-4 \cot (2) \csc (2)) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {2}{5} \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{40} e^2 \csc (2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5-\frac {2}{5} \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {1}{120} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5+\frac {32}{15} \csc ^4(2) (x-2)^5+\frac {2}{5} e^2 \csc ^3(2) (x-2)^5+\frac {1}{120} e^4 \csc ^2(2) (x-2)^5-\frac {1}{12} \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^4-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4+\frac {1}{2} \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4-\frac {8}{3} \csc ^3(2) (x-2)^4-\frac {1}{3} e^2 \csc ^2(2) (x-2)^4-\frac {1}{6} (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^3+\frac {8}{3} \csc ^2(2) (x-2)^3+\frac {1}{6} e^2 \csc (2) (x-2)^3-2 \csc (2) (x-2)^2+x-2\right )+c_1 \left (-\frac {1}{60} \left (-2 e^2 \csc (2)+4 e^2 \cot (2) \csc (2)-3 e^2 \cot ^2(2) \csc (2)+3 e^2 \cot ^3(2) \csc (2)\right ) (x-2)^5+\frac {1}{15} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{20} e^2 \csc (2) \left (3 \csc (2)-4 \cot (2) \csc (2)+4 \cot ^2(2) \csc (2)\right ) (x-2)^5-\frac {1}{40} (-4 \csc (2)+4 \cot (2) \csc (2)) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {2}{15} \csc ^2(2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{30} e^2 \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^5-\frac {1}{10} e^2 \csc ^2(2) (-4 \csc (2)+4 \cot (2) \csc (2)) (x-2)^5+\frac {1}{15} e^2 \csc ^2(2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^5-\frac {8}{15} e^2 \csc ^4(2) (x-2)^5-\frac {1}{15} e^4 \csc ^3(2) (x-2)^5-\frac {1}{12} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)-e^2 \cot ^2(2) \csc (2)\right ) (x-2)^4+\frac {1}{6} \csc (2) \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^4-\frac {1}{12} e^2 \csc (2) (4 \csc (2)-4 \cot (2) \csc (2)) (x-2)^4+\frac {2}{3} e^2 \csc ^3(2) (x-2)^4+\frac {1}{24} e^4 \csc ^2(2) (x-2)^4-\frac {1}{6} \left (-e^2 \csc (2)+e^2 \cot (2) \csc (2)\right ) (x-2)^3-\frac {2}{3} e^2 \csc ^2(2) (x-2)^3+\frac {1}{2} e^2 \csc (2) (x-2)^2+1\right )
\]