87.26.33 problem 45

Internal problem ID [23867]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 45
Date solved : Thursday, October 02, 2025 at 09:46:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{3}+\frac {21}{128} x^{4}+\frac {47}{512} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{8} x^{2}+\frac {9}{32} x^{3}+\frac {61}{512} x^{4}+\frac {1219}{10240} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (-\frac {77 x^5}{1536}-\frac {31 x^4}{384}-\frac {x^3}{8}-\frac {x^2}{2}+1\right )+c_2 \left (-\frac {343 x^5}{30720}-\frac {3 x^4}{512}-\frac {5 x^3}{96}+\frac {x^2}{8}+x\right ) \]

✓ Sympy. Time used: 0.385 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*(x + 2)*Derivative(y(x), (x, 2)) + (x + 1/2)*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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {31 x^{4}}{384} - \frac {x^{3}}{8} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {3 x^{3}}{512} - \frac {5 x^{2}}{96} + \frac {x}{8} + 1\right ) + O\left (x^{6}\right ) \]