56.4.62 problem 59

Internal problem ID [8951]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 59
Date solved : Wednesday, March 05, 2025 at 07:09:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y&=x \,{\mathrm e}^{x} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (\frac {x^5}{120}-\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (\frac {7 x^5}{120}-\frac {x^4}{12}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]

✗ Sympy

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

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