35.1.6 problem Ex. 6(v), page 257

Internal problem ID [8105]
Book : A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New York, reprinted 1956
Section : Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page 243
Problem number : Ex. 6(v), page 257
Date solved : Tuesday, September 30, 2025 at 05:15:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 38

Order:=6; 
ode:=(1-x)*x^2*diff(diff(y(x),x),x)+(5*x-4)*x*diff(y(x),x)+(6-9*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\left (x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 30

ode=(1-x)*x^2*D[y[x],{x,2}]+(5*x-4)*x*D[y[x],x]+(6-9*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x^3+c_1 \left (x^3 \log (x)-x^2 (3 x-1)\right ) \]

✓ Sympy. Time used: 0.406 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) + x*(5*x - 4)*Derivative(y(x), x) + (6 - 9*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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