87.15.9 problem 9

Internal problem ID [23557]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 115
Problem number : 9
Date solved : Friday, October 03, 2025 at 08:03:46 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime \prime }+2 y^{\prime }-x^{3} y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

Order:=6; 
ode:=diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)+2*diff(y(x),x)-x^3*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 1, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1+x -\frac {1}{3} x^{3}+\frac {1}{3} x^{4}-\frac {7}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 27

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

\[ y(x)\to -\frac {7 x^5}{30}+\frac {x^4}{3}-\frac {x^3}{3}+x+1 \]

✗ Sympy

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

Series solution not supported for ode of order > 2