14.26.27 problem 21

Internal problem ID [4052]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 07:01:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \left (\frac {1}{2} x^2 (5 x+4) \log (x)-\frac {3 x^4-6 x^3-6 x^2+4 x-4}{4 x}\right )+c_2 \left (\frac {x^6}{12}+\frac {7 x^5}{24}+\frac {3 x^4}{4}+\frac {5 x^3}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.298 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {7 x^{3}}{24} + \frac {3 x^{2}}{4} + \frac {5 x}{4} + 1\right ) + O\left (x^{6}\right ) \]