87.24.21 problem 21

Internal problem ID [23803]
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 : 21
Date solved : Thursday, October 02, 2025 at 09:45:21 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: 0.011 (sec). Leaf size: 42

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);
 

\[ y = c_1 \left (3+x \right )^{4} \left (1-\frac {3}{5} \left (3+x \right )+\frac {1}{5} \left (3+x \right )^{2}-\frac {1}{21} \left (3+x \right )^{3}+\frac {1}{112} \left (3+x \right )^{4}-\frac {1}{720} \left (3+x \right )^{5}+\operatorname {O}\left (\left (3+x \right )^{6}\right )\right )+c_2 \left (-144+48 \left (3+x \right )-2 \left (3+x \right )^{4}+\frac {6}{5} \left (3+x \right )^{5}+\operatorname {O}\left (\left (3+x \right )^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 72

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}]
 

\[ y(x)\to c_1 \left (\frac {1}{72} (x+3)^4+\frac {1}{3} (-x-3)+1\right )+c_2 \left (\frac {1}{112} (x+3)^8-\frac {1}{21} (x+3)^7+\frac {1}{5} (x+3)^6-\frac {3}{5} (x+3)^5+(x+3)^4\right ) \]

✓ Sympy. Time used: 0.301 (sec). Leaf size: 24

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_regular",x0=-3,n=6)
 

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