7.13.14 problem 14

Internal problem ID [413]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 14
Date solved : Thursday, March 13, 2025 at 03:35:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y&=x \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 49

Order:=6; 
ode:=diff(diff(y(x),x),x)+y(x) = x; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}-\frac {x^{5}}{120}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 56

ode=D[y[x],{x,2}]+y[x]==x; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✗ Sympy

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

ValueError : ODE -x + y(x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular