84.33.9 problem 20.10

Internal problem ID [22326]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Solved problems. Page 109
Problem number : 20.10
Date solved : Thursday, October 02, 2025 at 08:37:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }+\left (x^{3}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 64

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*exp(x)*diff(y(x),x)+(x^3-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{3} x +\frac {1}{48} x^{2}-\frac {43}{720} x^{3}+\frac {293}{11520} x^{4}-\frac {1}{1792} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{96} x^{4}+\frac {43}{1440} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {1}{2} x^{2}+\frac {2}{3} x^{3}-\frac {145}{384} x^{4}+\frac {4633}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 83

ode=x^2*D[y[x],{x,2}]+x*Exp[x]*D[y[x],x]+(x^3-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{192} x \left (x^2-16 x+48\right ) \log (x)+\frac {143 x^4-224 x^3+96 x^2-768 x+768}{768 x}\right )+c_2 \left (\frac {293 x^5}{11520}-\frac {43 x^4}{720}+\frac {x^3}{48}-\frac {x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.402 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*exp(x)*Derivative(y(x), x) + (x**3 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (1 - \frac {x^{3}}{15}\right ) + \frac {C_{1} \left (\frac {x^{6}}{72} - \frac {x^{3}}{3} + 1\right )}{x} + O\left (x^{6}\right ) \]