82.48.16 problem Ex. 16

Internal problem ID [18926]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 16
Date solved : Monday, March 31, 2025 at 06:25:10 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(diff(y(x),x),x),x)+cos(x)*diff(diff(y(x),x),x)-2*sin(x)*diff(y(x),x)-y(x)*cos(x) = sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_3 -\frac {\int \left (-8 c_1 x -4 c_2 +\sin \left (2 x \right )\right ) {\mathrm e}^{\sin \left (x \right )}d x}{4}\right ) {\mathrm e}^{-\sin \left (x \right )} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 75

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

\[ y(x)\to \exp \left (\int _1^x-\cos (K[3])dK[3]\right ) \left (\int _1^x\exp \left (-\int _1^{K[4]}-\cos (K[3])dK[3]\right ) \left (c_2+\int _1^{K[4]}\left (c_1+\int _1^{K[2]}\sin (2 K[1])dK[1]\right )dK[2]\right )dK[4]+c_3\right ) \]

✗ Sympy

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

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