87.26.29 problem 40

Internal problem ID [23863]
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 : 40
Date solved : Thursday, October 02, 2025 at 09:45:59 PM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 38

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

\[ y = \left (5 x -\frac {9}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{288} x^{4}+\frac {1}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 69

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

\[ y(x)\to c_1 \left (\frac {x^2}{2}-2 x+1\right )+c_2 \left (\frac {x^5}{3600}+\frac {x^4}{288}+\frac {x^3}{18}-\frac {9 x^2}{4}+\left (\frac {x^2}{2}-2 x+1\right ) \log (x)+5 x\right ) \]

✓ Sympy. Time used: 0.264 (sec). Leaf size: 17

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

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