40.16.10 problem 17

Internal problem ID [6801]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 25. Integration in series. Supplemetary problems. Page 205
Problem number : 17
Date solved : Sunday, March 30, 2025 at 11:23:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x^{2} y&=x^{2}+x +1 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 44

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

\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 49

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

\[ y(x)\to c_2 \left (x-\frac {x^5}{20}\right )+\frac {x^4}{12}+c_1 \left (1-\frac {x^4}{12}\right )+\frac {x^3}{6}+\frac {x^2}{2} \]

✗ Sympy

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

ValueError : ODE x**2*y(x) - x**2 - x + Derivative(y(x), (x, 2)) - 1 does not match hint 2nd_power_series_regular