36.5.8 problem 8

Internal problem ID [6352]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 8
Date solved : Sunday, March 30, 2025 at 10:53:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}-\frac {3}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {7}{24} x^{4}+\frac {23}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {3 x^5}{20}+\frac {x^4}{4}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {23 x^5}{120}-\frac {7 x^4}{24}+\frac {x^3}{3}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 1.006 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} e^{- 2 x}}{12} - \frac {x^{3} e^{- x}}{3} + 1\right ) + C_{1} x \left (- \frac {x^{3} e^{- x}}{12} - \frac {x^{3} e^{- 3 x}}{24} + \frac {x^{2} e^{- 2 x}}{6} - \frac {x e^{- x}}{2} + 1\right ) + O\left (x^{6}\right ) \]