35.1.2 problem Ex. 6(i), page 257

Internal problem ID [8101]
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(i), page 257
Date solved : Tuesday, September 30, 2025 at 05:15:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 2760

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

Too large to display

✓ Sympy. Time used: 0.323 (sec). Leaf size: 8

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

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