83.14.1 problem 1

Internal problem ID [22016]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter X. Solution in power series. Ex. XVIII at page 174
Problem number : 1
Date solved : Friday, October 03, 2025 at 07:59:45 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

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

\[ y = 1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 19

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

\[ y(x)\to \frac {x^4}{24}-\frac {x^2}{2}+1 \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + y(t) - exp(t) - Derivative(x(t), (t, 2)) + Derivative(y(t), t),0),Eq(x(t) - Derivative(x(t), t) + Derivative(y(t), t) - exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Series solution not supported for ode of order > 2