9.16.16 problem 16

Internal problem ID [3236]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 25, page 112
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 06:30:05 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

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

\[ y = c_2 \,x^{-\frac {\sqrt {5}}{2}+\frac {1}{2}}+c_3 \,x^{\frac {\sqrt {5}}{2}+\frac {1}{2}}+\left (-\frac {1}{85}+\frac {9 i}{170}\right ) x^{-i}+\left (-\frac {1}{85}-\frac {9 i}{170}\right ) x^{i}+c_1 \,x^{4} \]

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to c_2 x^{\frac {1}{2} \left (1+\sqrt {5}\right )}+c_1 x^{\frac {1}{2}-\frac {\sqrt {5}}{2}}+c_3 x^4+\frac {9}{85} \sin (\log (x))-\frac {2}{85} \cos (\log (x)) \end{align*}

✓ Sympy. Time used: 0.602 (sec). Leaf size: 61

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

\[ y{\left (x \right )} = \frac {C_{1} + \frac {x^{- \frac {1}{2} + \frac {\sqrt {5}}{2}} \left (C_{2} x^{4} + C_{3} x^{\frac {1}{2} + \frac {\sqrt {5}}{2}} + 9 \sin {\left (\log {\left (x \right )} \right )} - 2 \cos {\left (\log {\left (x \right )} \right )}\right )}{85}}{x^{- \frac {1}{2} + \frac {\sqrt {5}}{2}}} \]