43.17.10 problem 2(d)

Internal problem ID [9003]
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)
Date solved : Tuesday, September 30, 2025 at 06:01:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (-3 x^{2}+x \right ) y^{\prime }+{\mathrm e}^{x} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 55

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+(-3*x^2+x)*diff(y(x),x)+exp(x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{-i} \left (1+\left (1-i\right ) x +\left (\frac {7}{16}-\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}-\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}-\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}-\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}-\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}-\frac {3271304833 i}{812078991360}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{i} \left (1+\left (1+i\right ) x +\left (\frac {7}{16}+\frac {13 i}{16}\right ) x^{2}+\left (\frac {7}{39}+\frac {395 i}{936}\right ) x^{3}+\left (\frac {2117}{29952}+\frac {5197 i}{29952}\right ) x^{4}+\left (\frac {5521}{217152}+\frac {642043 i}{10857600}\right ) x^{5}+\left (\frac {782461}{97718400}+\frac {8813057 i}{521164800}\right ) x^{6}+\left (\frac {1238071931}{580056422400}+\frac {3271304833 i}{812078991360}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 122

ode=x^2*D[y[x],{x,2}]+(x-3*x^2)*D[y[x],x]+Exp[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \left (\frac {1}{97718400}+\frac {11 i}{1563494400}\right ) c_1 x^i \left ((1302761+756800 i) x^6+(4384656+2763936 i) x^5+(12605400+8289000 i) x^4+(31161600+19814400 i) x^3+(66096000+33955200 i) x^2+(111974400+20736000 i) x+(66355200-45619200 i)\right )-\left (\frac {11}{1563494400}+\frac {i}{97718400}\right ) c_2 x^{-i} \left ((756800+1302761 i) x^6+(2763936+4384656 i) x^5+(8289000+12605400 i) x^4+(19814400+31161600 i) x^3+(33955200+66096000 i) x^2+(20736000+111974400 i) x-(45619200-66355200 i)\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (-3*x**2 + x)*Derivative(y(x), x) + y(x)*exp(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : Expected Expr or iterable but got None