30.14.4 problem 4

Internal problem ID [7653]
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 : 4
Date solved : Tuesday, September 30, 2025 at 04:55:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-5 x +6\right ) y^{\prime \prime }-3 x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 54

Order:=6; 
ode:=(x^2-5*x+6)*diff(diff(y(x),x),x)-3*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{12} x^{2}+\frac {5}{216} x^{3}+\frac {5}{324} x^{4}+\frac {11}{1296} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{9} x^{3}+\frac {5}{108} x^{4}+\frac {29}{1080} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

ode=(x^2-5*x+6)*D[y[x],{x,2}]-3*x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {29 x^5}{1080}+\frac {5 x^4}{108}+\frac {x^3}{9}+x\right )+c_1 \left (\frac {11 x^5}{1296}+\frac {5 x^4}{324}+\frac {5 x^3}{216}+\frac {x^2}{12}+1\right ) \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + (x**2 - 5*x + 6)*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {5 x^{4}}{324} + \frac {5 x^{3}}{216} + \frac {x^{2}}{12} + 1\right ) + C_{1} x \left (\frac {5 x^{3}}{108} + \frac {x^{2}}{9} + 1\right ) + O\left (x^{6}\right ) \]