87.24.1 problem 1

Internal problem ID [23783]
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 218
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:45:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 58

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

\[ y = c_1 \,x^{2} \left (1+x +\frac {5}{8} x^{2}+\frac {7}{24} x^{3}+\frac {7}{64} x^{4}+\frac {11}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{2}-x^{3}-\frac {5}{8} x^{4}-\frac {7}{24} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x +\frac {2}{3} x^{3}+\frac {61}{96} x^{4}+\frac {59}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

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

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

\[ y(x)\to c_1 \left (-\frac {3 x^5}{40}-\frac {x^4}{6}-\frac {x^3}{6}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{8}+\frac {5 x^4}{24}+\frac {x^3}{3}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.271 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(2*x + 1)*Derivative(y(x), x) + y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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