62.31.4 problem Ex 4

Internal problem ID [12905]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 4
Date solved : Monday, March 31, 2025 at 07:24:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 82

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

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

✗ Sympy

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

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