20.24.19 problem Problem 20

Internal problem ID [4004]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 20
Date solved : Sunday, March 30, 2025 at 02:14:16 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 45

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), x) + x*y(x) - 2*cos(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 2*x**2*Derivative(y(x), x) + x*y(x) - 2*cos(x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular