87.25.22 problem 22

Internal problem ID [23825]
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 232
Problem number : 22
Date solved : Thursday, October 02, 2025 at 09:45:33 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }-2 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

With initial conditions

\begin{align*} y \left (3\right )&=1 \\ y^{\prime }\left (3\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-2*diff(y(x),x)+x*y(x) = 0; 
ic:=[y(3) = 1, D(y)(3) = -2]; 
dsolve([ode,op(ic)],y(x),type='series',x=3);
 

\[ y = 1-2 \left (x -3\right )-\frac {7}{6} \left (x -3\right )^{2}+\frac {4}{27} \left (x -3\right )^{3}+\frac {11}{72} \left (x -3\right )^{4}+\frac {1}{540} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to \frac {1}{540} (x-3)^5+\frac {11}{72} (x-3)^4+\frac {4}{27} (x-3)^3-\frac {7}{6} (x-3)^2-2 (x-3)+1 \]

✓ Sympy. Time used: 0.283 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 3\right )^{4}}{18} - \frac {7 \left (x - 3\right )^{3}}{54} + \frac {\left (x - 3\right )^{2}}{3} - 3\right ) + C_{1} \left (\frac {\left (x - 3\right )^{4}}{24} - \frac {\left (x - 3\right )^{3}}{9} - \frac {\left (x - 3\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]