49.18.3 problem 2
Internal
problem
ID
[7720]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
4.
Linear
equations
with
Regular
Singular
Points.
Page
159
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 03:10:34 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 55
Order:=8;
dsolve(x^2*diff(y(x),x$2)+x*exp(x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[
y = c_{1} x^{-i} \left (1+\left (-\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {3}{80}+\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}-\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}-\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}+\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}+\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}+\frac {8634893 i}{580056422400}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {3}{80}-\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}+\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}+\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}-\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}-\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}-\frac {8634893 i}{580056422400}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*Exp[x]*D[y[x],x]+y[x]==0,y[x],{x,0,"8"-1}]
\[
y(x)\to \left (\frac {11}{1563494400}+\frac {i}{97718400}\right ) c_2 x^{-i} \left ((4913+7070 i) x^6-(8568-32328 i) x^5-(132840+24120 i) x^4-(247680+869760 i) x^3+(2540160-1918080 i) x^2-(4976640-35665920 i) x+(45619200-66355200 i)\right )-\left (\frac {1}{97718400}+\frac {11 i}{1563494400}\right ) c_1 x^i \left ((7070+4913 i) x^6+(32328-8568 i) x^5-(24120+132840 i) x^4-(869760+247680 i) x^3-(1918080-2540160 i) x^2+(35665920-4976640 i) x-(66355200-45619200 i)\right )
\]