54.3.17 problem 17

Internal problem ID [8573]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 17
Date solved : Sunday, March 30, 2025 at 01:20:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 59

Order:=8; 
ode:=diff(diff(y(x),x),x)+x*diff(y(x),x)+3*y(x) = x^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{2} x^{2}+\frac {5}{8} x^{4}-\frac {7}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {1}{5} x^{5}-\frac {4}{105} x^{7}\right ) y^{\prime }\left (0\right )+\frac {x^{4}}{12}-\frac {7 x^{6}}{360}+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 70

ode=D[y[x],{x,2}]+x*D[y[x],x]+3*y[x]==x^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {7 x^6}{360}+\frac {x^4}{12}+c_2 \left (-\frac {4 x^7}{105}+\frac {x^5}{5}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {7 x^6}{48}+\frac {5 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + x*Derivative(y(x), x) + 3*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE -x**2 + x*Derivative(y(x), x) + 3*y(x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular