30.14.2 problem 2

Internal problem ID [7651]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 04:55:29 PM
CAS classification : [_Hermite]

\begin{align*} y^{\prime \prime }-x y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 72

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

\[ y = \left (1+\frac {3 \left (x -2\right )^{2}}{2}+\left (x -2\right )^{3}+\frac {9 \left (x -2\right )^{4}}{8}+\frac {3 \left (x -2\right )^{5}}{4}\right ) y \left (2\right )+\left (x -2+\left (x -2\right )^{2}+\frac {4 \left (x -2\right )^{3}}{3}+\frac {13 \left (x -2\right )^{4}}{12}+\frac {5 \left (x -2\right )^{5}}{6}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 79

ode=D[y[x],{x,2}]-x*D[y[x],x]-3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (\frac {3}{4} (x-2)^5+\frac {9}{8} (x-2)^4+(x-2)^3+\frac {3}{2} (x-2)^2+1\right )+c_2 \left (\frac {5}{6} (x-2)^5+\frac {13}{12} (x-2)^4+\frac {4}{3} (x-2)^3+(x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (x + \frac {13 \left (x - 2\right )^{4}}{12} + \frac {4 \left (x - 2\right )^{3}}{3} + \left (x - 2\right )^{2} - 2\right ) + C_{1} \left (\frac {9 \left (x - 2\right )^{4}}{8} + \left (x - 2\right )^{3} + \frac {3 \left (x - 2\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]