20.26.1 problem Example 11.5.2 page 763

Internal problem ID [4026]
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 : Example 11.5.2 page 763
Date solved : Sunday, March 30, 2025 at 02:14:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (3+x \right ) y^{\prime }+\left (4-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*(x+3)*diff(y(x),x)+(4-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+3 x +3 x^{2}+\frac {5}{3} x^{3}+\frac {5}{8} x^{4}+\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-5\right ) x -\frac {29}{4} x^{2}-\frac {173}{36} x^{3}-\frac {193}{96} x^{4}-\frac {1459}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{2} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 118

ode=x^2*D[y[x],{x,2}]-x*(3+x)*D[y[x],x]+(4-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {7 x^5}{40}+\frac {5 x^4}{8}+\frac {5 x^3}{3}+3 x^2+3 x+1\right ) x^2+c_2 \left (\left (-\frac {1459 x^5}{2400}-\frac {193 x^4}{96}-\frac {173 x^3}{36}-\frac {29 x^2}{4}-5 x\right ) x^2+\left (\frac {7 x^5}{40}+\frac {5 x^4}{8}+\frac {5 x^3}{3}+3 x^2+3 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.974 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(x + 3)*Derivative(y(x), x) + (4 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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