85.72.16 problem 2 (f)

Internal problem ID [22952]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 316
Problem number : 2 (f)
Date solved : Thursday, October 02, 2025 at 09:16:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 69

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

\[ y = \left (1+\frac {\left (x -1\right )^{2}}{4}-\frac {5 \left (x -1\right )^{3}}{24}+\frac {7 \left (x -1\right )^{4}}{48}-\frac {107 \left (x -1\right )^{5}}{960}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{12}-\frac {5 \left (x -1\right )^{4}}{48}+\frac {\left (x -1\right )^{5}}{12}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (-\frac {107}{960} (x-1)^5+\frac {7}{48} (x-1)^4-\frac {5}{24} (x-1)^3+\frac {1}{4} (x-1)^2+1\right )+c_2 \left (\frac {1}{12} (x-1)^5-\frac {5}{48} (x-1)^4+\frac {1}{12} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.312 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{2} \left (\frac {7 \left (x - 1\right )^{4}}{48} - \frac {5 \left (x - 1\right )^{3}}{24} + \frac {\left (x - 1\right )^{2}}{4} + 1\right ) + C_{1} \left (x - \frac {5 \left (x - 1\right )^{4}}{48} + \frac {\left (x - 1\right )^{3}}{12} - 1\right ) + O\left (x^{6}\right ) \]