87.25.13 problem 13

Internal problem ID [23816]
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 : 13
Date solved : Thursday, October 02, 2025 at 09:45:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime }-3 y^{\prime }+\left (x -1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-20 \\ y^{\prime }\left (1\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 20

Order:=6; 
ode:=(x^2+2)*diff(diff(y(x),x),x)-3*diff(y(x),x)+(x-1)*y(x) = 0; 
ic:=[y(1) = -20, D(y)(1) = -2]; 
dsolve([ode,op(ic)],y(x),type='series',x=1);
 

\[ y = -20-2 \left (x -1\right )-\left (x -1\right )^{2}+\left (x -1\right )^{3}+\frac {1}{36} \left (x -1\right )^{4}-\frac {4}{45} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

ode=(x^2+2)*D[y[x],{x,2}]-3*D[y[x],x]+(x-1)*y[x]==0; 
ic={y[1]==-20,Derivative[1][y][1] ==-2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to -\frac {4}{45} (x-1)^5+\frac {1}{36} (x-1)^4+(x-1)^3-(x-1)^2-2 (x-1)-20 \]

✓ Sympy. Time used: 0.320 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*y(x) + (x**2 + 2)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x),0) 
ics = {y(1): -20, Subs(Derivative(y(x), x), x, 1): -2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {13 \left (x - 1\right )^{4}}{216} + \frac {\left (x - 1\right )^{3}}{18} + \frac {\left (x - 1\right )^{2}}{2} - 1\right ) + C_{1} \left (\frac {\left (x - 1\right )^{4}}{216} - \frac {\left (x - 1\right )^{3}}{18} + 1\right ) + O\left (x^{6}\right ) \]