87.26.13 problem 21

Internal problem ID [23847]
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 : 21
Date solved : Thursday, October 02, 2025 at 09:45:47 PM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 40

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (7 x -\frac {23}{4} x^{2}+\frac {11}{12} x^{3}-\frac {1}{96} x^{4}-\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-3 x +\frac {3}{2} x^{2}-\frac {1}{6} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 83

ode=x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^3}{6}+\frac {3 x^2}{2}-3 x+1\right )+c_2 \left (-\frac {x^5}{2400}-\frac {x^4}{96}+\frac {11 x^3}{12}-\frac {23 x^2}{4}+\left (-\frac {x^3}{6}+\frac {3 x^2}{2}-3 x+1\right ) \log (x)+7 x\right ) \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (1 - x)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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