87.16.21 problem 21

Internal problem ID [23590]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 119
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:43:16 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} 3 x y^{\prime \prime \prime }+5 y^{\prime \prime }+6 y^{\prime }&=3 \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 98

ode:=3*x*diff(diff(diff(y(x),x),x),x)+5*diff(diff(y(x),x),x)+6*diff(y(x),x) = 3*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \int \frac {-\pi \int x^{{1}/{3}} \operatorname {BesselY}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right ) \cos \left (x \right )d x \operatorname {BesselJ}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right )+\pi \int x^{{1}/{3}} \operatorname {BesselJ}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right ) \cos \left (x \right )d x \operatorname {BesselY}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right )+\operatorname {BesselJ}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right ) c_2 +\operatorname {BesselY}\left (\frac {2}{3}, 2 \sqrt {2}\, \sqrt {x}\right ) c_1}{x^{{1}/{3}}}d x +c_3 \]

✓ Mathematica. Time used: 87.569 (sec). Leaf size: 210

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

\begin{align*} y(x)&\to \int _1^x\frac {\operatorname {BesselJ}\left (-\frac {2}{3},2 \sqrt {2} \sqrt {K[3]}\right ) \operatorname {Gamma}\left (\frac {1}{3}\right ) c_1+\operatorname {BesselJ}\left (-\frac {2}{3},2 \sqrt {2} \sqrt {K[3]}\right ) \operatorname {Gamma}\left (\frac {1}{3}\right ) \int _1^{K[3]}-\frac {3 \operatorname {BesselJ}\left (\frac {2}{3},2 \sqrt {2} \sqrt {K[1]}\right ) \cos (K[1]) \operatorname {Gamma}\left (\frac {5}{3}\right ) \sqrt [3]{K[1]}}{2^{2/3}}dK[1]+\operatorname {BesselJ}\left (\frac {2}{3},2 \sqrt {2} \sqrt {K[3]}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) \int _1^{K[3]}\frac {3 \operatorname {BesselJ}\left (-\frac {2}{3},2 \sqrt {2} \sqrt {K[2]}\right ) \cos (K[2]) \operatorname {Gamma}\left (\frac {1}{3}\right ) \sqrt [3]{K[2]}}{2^{2/3}}dK[2]+\operatorname {BesselJ}\left (\frac {2}{3},2 \sqrt {2} \sqrt {K[3]}\right ) c_2 \operatorname {Gamma}\left (\frac {5}{3}\right )}{\sqrt [3]{2} \sqrt [3]{K[3]}}dK[3]+c_3 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE x*Derivative(y(x), (x, 3))/2 - cos(x)/2 + Derivative(y(x), x) +