Internal problem ID [5299]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 85
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);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (-\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {3}{80}+\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}-\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}-\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}+\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}+\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}+\frac {8634893 i}{580056422400}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {3}{80}-\frac {i}{80}\right ) x^{2}+\left (\frac {67}{9360}+\frac {9 i}{1040}\right ) x^{3}+\left (-\frac {103}{149760}+\frac {229 i}{149760}\right ) x^{4}+\left (-\frac {2831}{7238400}-\frac {607 i}{4343040}\right ) x^{5}+\left (-\frac {59077}{1563494400}-\frac {26063 i}{260582400}\right ) x^{6}+\left (\frac {22952047}{2030197478400}-\frac {8634893 i}{580056422400}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*y''[x]+x*Exp[x]*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \left (\frac {11}{1563494400}+\frac {i}{97718400}\right ) c_2 x^{-i} \left ((4913+7070 i) x^6-(8568-32328 i) x^5-(132840+24120 i) x^4-(247680+869760 i) x^3+(2540160-1918080 i) x^2-(4976640-35665920 i) x+(45619200-66355200 i)\right )-\left (\frac {1}{97718400}+\frac {11 i}{1563494400}\right ) c_1 x^i \left ((7070+4913 i) x^6+(32328-8568 i) x^5-(24120+132840 i) x^4-(869760+247680 i) x^3-(1918080-2540160 i) x^2+(35665920-4976640 i) x-(66355200-45619200 i)\right ) \]