58.2.18 problem 19

Internal problem ID [9141]
Book : Second order enumerated odes
Section : section 2
Problem number : 19
Date solved : Sunday, March 30, 2025 at 02:23:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+a^{2} y&=\frac {1}{x^{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

ode:=x^6*diff(diff(y(x),x),x)+3*x^5*diff(y(x),x)+a^2*y(x) = 1/x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (\frac {a}{2 x^{2}}\right ) c_2 +\cos \left (\frac {a}{2 x^{2}}\right ) c_1 +\frac {1}{a^{2} x^{2}} \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 104

ode=x^6*D[y[x],{x,2}]+3*x^5*D[y[x],x]+a^2*y[x]==1/x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\sin \left (\frac {a}{2 x^2}\right ) \int _1^x\frac {\cos \left (\frac {a}{2 K[1]^2}\right )}{a K[1]^5}dK[1]+\frac {-2 a^3 c_2 x^2 \sin \left (\frac {a}{2 x^2}\right )-2 x^2 \sin \left (\frac {a}{x^2}\right )+a \cos \left (\frac {a}{x^2}\right )+a}{2 a^3 x^2}+c_1 \cos \left (\frac {a}{2 x^2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*y(x) + x**6*Derivative(y(x), (x, 2)) + 3*x**5*Derivative(y(x), x) - 1/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a**2*x**2*y(x) - x**8*Derivative(y(x), (x, 2)) + 1)/(3*x**7) cannot be solved by the factorable group method