87.26.21 problem 30

Internal problem ID [23855]
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 253
Problem number : 30
Date solved : Thursday, October 02, 2025 at 09:45:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 66

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

\[ y = \frac {c_1 \left (x +1\right )^{3} \left (1+\frac {1}{4} \left (x +1\right )+\frac {1}{40} \left (x +1\right )^{2}+\frac {1}{720} \left (x +1\right )^{3}+\frac {1}{20160} \left (x +1\right )^{4}+\frac {1}{806400} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )+c_2 \left (\ln \left (x +1\right ) \left (\left (x +1\right )^{3}+\frac {1}{4} \left (x +1\right )^{4}+\frac {1}{40} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )+\left (12-6 \left (x +1\right )+3 \left (x +1\right )^{2}-\frac {5}{16} \left (x +1\right )^{4}-\frac {39}{800} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )\right )}{x +1} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 106

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

\[ y(x)\to c_2 \left (\frac {(x+1)^6}{20160}+\frac {1}{720} (x+1)^5+\frac {1}{40} (x+1)^4+\frac {1}{4} (x+1)^3+(x+1)^2\right )+c_1 \left (\frac {1}{48} (x+1)^2 (x+5) \log (x+1)-\frac {19 (x+1)^4+16 (x+1)^3-144 (x+1)^2+288 (x+1)-576}{576 (x+1)}\right ) \]

✓ Sympy. Time used: 0.337 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \frac {C_{1} \left (x + 1\right )^{2} \left (180 x + \left (x + 1\right )^{3} + 18 \left (x + 1\right )^{2} + 900\right )}{720} + O\left (x^{6}\right ) \]