56.4.27 problem 24

Internal problem ID [8916]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 24
Date solved : Sunday, March 30, 2025 at 01:53:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y&=x^{3} \cos \left (x \right )+\sin \left (x \right )^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 47

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(-x^2+1)*y(x) = x^3*cos(x)+sin(x)^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{10} x -\frac {1}{90} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Mathematica. Time used: 0.491 (sec). Leaf size: 199

ode=2*x^2*D[y[x],{x,2}]-x*D[y[x],x]+(1-x^2)*y[x]==x^3*Cos[x]+Sin[x]^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{396}+\frac {4 x^{9/2}}{45}+\frac {x^{7/2}}{15}-\frac {2 x^{5/2}}{5}-\frac {2 x^{3/2}}{3}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (-\frac {x^6}{168}-\frac {13 x^5}{12600}-\frac {x^4}{12}-\frac {x^3}{18}+\frac {x^2}{2}+x\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*cos(x) + 2*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (1 - x**2)*y(x) - sin(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -x**3*cos(x) + 2*x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (1 - x**2)*y(x) - sin(x)**2 does not match hint 2nd_power_series_regular