43.17.6 problem 1(f)

Internal problem ID [8999]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(f)
Date solved : Tuesday, September 30, 2025 at 06:01:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -2 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 60

Order:=8; 
ode:=(x^2+x-2)^2*diff(diff(y(x),x),x)+3*(x+2)*diff(y(x),x)+(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=-2);
 

\[ y = \frac {c_1 \left (1-\frac {5}{9} \left (x +2\right )+\frac {23}{324} \left (x +2\right )^{2}+\frac {271}{43740} \left (x +2\right )^{3}+\frac {10517}{12597120} \left (x +2\right )^{4}+\frac {778801}{6235574400} \left (x +2\right )^{5}+\frac {16965493}{942818849280} \left (x +2\right )^{6}+\frac {899971067}{458981357990400} \left (x +2\right )^{7}+\operatorname {O}\left (\left (x +2\right )^{8}\right )\right )+c_2 \left (x +2\right )^{{4}/{3}} \left (1-\frac {1}{21} \left (x +2\right )-\frac {11}{1260} \left (x +2\right )^{2}-\frac {53}{29484} \left (x +2\right )^{3}-\frac {11093}{28304640} \left (x +2\right )^{4}-\frac {709507}{8066822400} \left (x +2\right )^{5}-\frac {5797423}{290405606400} \left (x +2\right )^{6}-\frac {52991201}{11727918720000} \left (x +2\right )^{7}+\operatorname {O}\left (\left (x +2\right )^{8}\right )\right )}{\left (x +2\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 148

ode=(x^2+x-2)^2*D[y[x],{x,2}]+3*(x+2)*D[y[x],x]+(x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-2,7}]
 

\[ y(x)\to c_1 (x+2) \left (-\frac {52991201 (x+2)^7}{11727918720000}-\frac {5797423 (x+2)^6}{290405606400}-\frac {709507 (x+2)^5}{8066822400}-\frac {11093 (x+2)^4}{28304640}-\frac {53 (x+2)^3}{29484}-\frac {11 (x+2)^2}{1260}+\frac {1}{21} (-x-2)+1\right )+\frac {c_2 \left (\frac {899971067 (x+2)^7}{458981357990400}+\frac {16965493 (x+2)^6}{942818849280}+\frac {778801 (x+2)^5}{6235574400}+\frac {10517 (x+2)^4}{12597120}+\frac {271 (x+2)^3}{43740}+\frac {23}{324} (x+2)^2-\frac {5 (x+2)}{9}+1\right )}{\sqrt [3]{x+2}} \]

✓ Sympy. Time used: 42.899 (sec). Leaf size: 4605

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)*y(x) + (3*x + 6)*Derivative(y(x), x) + (x**2 + x - 2)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=-2,n=8)
 

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