87.24.20 problem 20

Internal problem ID [23802]
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 218
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:45:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 76

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

\[ y = \left (1+\frac {\left (x -3\right )^{2}}{12}-\frac {\left (x -3\right )^{3}}{54}+\frac {7 \left (x -3\right )^{4}}{2592}-\frac {\left (x -3\right )^{5}}{4320}\right ) y \left (3\right )+\left (x -3-\frac {\left (x -3\right )^{2}}{4}+\frac {\left (x -3\right )^{3}}{18}-\frac {7 \left (x -3\right )^{4}}{864}+\frac {\left (x -3\right )^{5}}{1440}\right ) y^{\prime }\left (3\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (-\frac {(x-3)^5}{4320}+\frac {7 (x-3)^4}{2592}-\frac {1}{54} (x-3)^3+\frac {1}{12} (x-3)^2+1\right )+c_2 \left (\frac {(x-3)^5}{1440}-\frac {7}{864} (x-3)^4+\frac {1}{18} (x-3)^3-\frac {1}{4} (x-3)^2+x-3\right ) \]

✓ Sympy. Time used: 0.286 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {7 \left (x - 3\right )^{4}}{864} + \frac {\left (x - 3\right )^{3}}{18} - \frac {\left (x - 3\right )^{2}}{4} - 3\right ) + C_{1} \left (\frac {7 \left (x - 3\right )^{4}}{2592} - \frac {\left (x - 3\right )^{3}}{54} + \frac {\left (x - 3\right )^{2}}{12} + 1\right ) + O\left (x^{6}\right ) \]