87.26.28 problem 39

Internal problem ID [23862]
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 : 39
Date solved : Thursday, October 02, 2025 at 09:45:58 PM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 36

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

\[ y = \left (3 x -\frac {1}{4} x^{2}-\frac {1}{36} x^{3}-\frac {1}{288} x^{4}-\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \left (-\frac {x^5}{2400}-\frac {x^4}{288}-\frac {x^3}{36}-\frac {x^2}{4}+3 x+(1-x) \log (x)\right )+c_1 (1-x) \]

✓ Sympy. Time used: 0.242 (sec). Leaf size: 10

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

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