34.13.10 problem 31

Internal problem ID [8051]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 18. Linear equations with variable coefficients (Equations of second order). Supplemetary problems. Page 120
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 05:14:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{8} y^{\prime \prime }+4 x^{7} y^{\prime }+y&=\frac {1}{x^{3}} \end{align*}

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

ode:=x^8*diff(diff(y(x),x),x)+4*x^7*diff(y(x),x)+y(x) = 1/x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (\frac {1}{3 x^{3}}\right ) c_2 +\cos \left (\frac {1}{3 x^{3}}\right ) c_1 +\frac {1}{x^{3}} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 90

ode=x^8*D[y[x],{x,2}]+4*x^7*D[y[x],x]+y[x]==1/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (-2 \sin \left (\frac {1}{3 x^3}\right ) \int _1^x\frac {\cos \left (\frac {1}{3 K[1]^3}\right )}{K[1]^7}dK[1]+\frac {1}{x^3}-3 \sin \left (\frac {2}{3 x^3}\right )+\frac {\cos \left (\frac {2}{3 x^3}\right )}{x^3}+2 c_1 \cos \left (\frac {1}{3 x^3}\right )-2 c_2 \sin \left (\frac {1}{3 x^3}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**8*Derivative(y(x), (x, 2)) + 4*x**7*Derivative(y(x), x) + y(x) - 1/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**11*Derivative(y(x), (x, 2)) - x**3*y(x) + 1)/(4*x**10) cannot be solved by the factorable group method