Internal problem ID [5296]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 2(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (-3 x^{2}+x \right ) y^{\prime }+y \,{\mathrm e}^{x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 85
Order:=8; dsolve(x^2*diff(y(x),x$2)+(x-3*x^2)*diff(y(x),x)+exp(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (1-i\right ) x +\left (\frac {7}{16}-\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}-\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}-\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}-\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}-\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}-\frac {3271304833 i}{812078991360}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (1+i\right ) x +\left (\frac {7}{16}+\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}+\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}+\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}+\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}+\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}+\frac {3271304833 i}{812078991360}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.075 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*y''[x]+(x-3*x^2)*y'[x]+Exp[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \left (\frac {1}{97718400}+\frac {11 i}{1563494400}\right ) c_1 x^i \left ((1302761+756800 i) x^6+(4384656+2763936 i) x^5+(12605400+8289000 i) x^4+(31161600+19814400 i) x^3+(66096000+33955200 i) x^2+(111974400+20736000 i) x+(66355200-45619200 i)\right )-\left (\frac {11}{1563494400}+\frac {i}{97718400}\right ) c_2 x^{-i} \left ((756800+1302761 i) x^6+(2763936+4384656 i) x^5+(8289000+12605400 i) x^4+(19814400+31161600 i) x^3+(33955200+66096000 i) x^2+(20736000+111974400 i) x-(45619200-66355200 i)\right ) \]