41.1.4 problem Ex. 6(iii), page 257

Internal problem ID [6817]
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(iii), page 257
Date solved : Sunday, March 30, 2025 at 11:23:36 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 46

Order:=6; 
ode:=x^3*(x^2+1)*diff(diff(diff(y(x),x),x),x)-(4*x^2+2)*x^2*diff(diff(y(x),x),x)+(10*x^2+4)*x*diff(y(x),x)-(12*x^2+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x \left (c_3 \left (2+2 x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (2+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (5+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \right ) \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 30

ode=x^3*(1+x^2)*D[y[x],{x,3}]-(2+4*x^2)*x^2*D[y[x],{x,2}]+(4+10*x^2)*x*D[y[x],x]-(4+12*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (2 x^3+2 x\right )+c_2 x^2+c_3 x^2 \log (x) \]

✗ Sympy

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

Series solution not supported for ode of order > 2