35.1.1 problem Ex. 5, page 256

Internal problem ID [8100]
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. 5, page 256
Date solved : Tuesday, September 30, 2025 at 05:15:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2+2 x +4 x^{2}+4 x^{3}+2 x^{4}+\frac {2}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (-\frac {5 x^4}{4}-\frac {5 x^3}{2}-\frac {5 x^2}{2}-x+1\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]

✓ Sympy. Time used: 0.447 (sec). Leaf size: 12

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

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