41.1.3 problem Ex. 6(ii), page 257

Internal problem ID [6816]
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(ii), page 257
Date solved : Sunday, March 30, 2025 at 11:23:35 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 71

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

\[ y = x \left (\left (2 x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )^{2} c_3 +\left (2+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 x +2 \left (\left (-4\right ) x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_3 +c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (5+\operatorname {O}\left (x^{6}\right )\right ) c_2 x +\left (2+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) c_3 \right ) \]

✓ Mathematica. Time used: 0.487 (sec). Leaf size: 49

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

\[ y(x)\to c_2 x^2+c_1 \left (2 \left (x^2+11 x+1\right ) x+2 x^2 \log ^2(x)-14 x^2 \log (x)\right )+c_3 x^2 \log (x) \]

✗ 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