81.12.21 problem 16-22

Internal problem ID [21674]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 16. Variation of Parameters. Page 375.
Problem number : 16-22
Date solved : Thursday, October 02, 2025 at 07:59:46 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+8 x y^{\prime }-8 y&=4 \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

ode:=x^3*diff(diff(diff(y(x),x),x),x)-4*x^2*diff(diff(y(x),x),x)+8*x*diff(y(x),x)-8*y(x) = 4*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\ln \left (x \right )}{2}-\frac {7}{8}+c_3 \,x^{4}+c_2 \,x^{2}+c_1 x \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to c_3 x^4+c_2 x^2-\frac {\log (x)}{2}+c_1 x-\frac {7}{8} \end{align*}

✓ Sympy. Time used: 0.268 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} x + C_{2} x^{2} + C_{3} x^{4} - \frac {\log {\left (x \right )}}{2} - \frac {7}{8} \]