69.24.3 problem 741

Internal problem ID [18502]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 741
Date solved : Thursday, October 02, 2025 at 03:14:25 PM
CAS classification : [_Jacobi]

\begin{align*} 9 x \left (1-x \right ) y^{\prime \prime }-12 y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

Order:=6; 
ode:=9*x*(1-x)*diff(diff(y(x),x),x)-12*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{7}/{3}} \left (1+\frac {4}{5} x +\frac {44}{65} x^{2}+\frac {77}{130} x^{3}+\frac {1309}{2470} x^{4}+\frac {119}{247} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{3} x +\frac {2}{9} x^{2}+\frac {14}{81} x^{3}+\frac {35}{243} x^{4}+\frac {91}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 85

ode=9*x*(1-x)*D[y[x],{x,2}]-12*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {91 x^5}{729}+\frac {35 x^4}{243}+\frac {14 x^3}{81}+\frac {2 x^2}{9}+\frac {x}{3}+1\right )+c_1 \left (\frac {119 x^5}{247}+\frac {1309 x^4}{2470}+\frac {77 x^3}{130}+\frac {44 x^2}{65}+\frac {4 x}{5}+1\right ) x^{7/3} \]

✓ Sympy. Time used: 0.380 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x*(1 - x)*Derivative(y(x), (x, 2)) + 4*y(x) - 12*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {162 x^{5}}{25} + \frac {108 x^{4}}{5} + 36 x^{3} + 18 x^{2} - 3 x + 1\right ) + C_{1} x^{\frac {7}{3}} \left (\frac {36 x^{2}}{65} + \frac {6 x}{5} + 1\right ) + O\left (x^{6}\right ) \]