36.6.21 problem 24

Internal problem ID [6382]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 24
Date solved : Sunday, March 30, 2025 at 10:54:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+3 y&=x^{2} \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)-2*x*diff(y(x),x)+3*y(x) = x^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{2} x^{2}-\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{4}}{12}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 49

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

\[ y(x)\to \frac {x^4}{12}+c_2 \left (-\frac {x^5}{40}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^4}{8}-\frac {3 x^2}{2}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - 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=6)
 

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