23.5.41 problem 41

Internal problem ID [6650]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 03:50:31 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 71

ode:=2*a^2*y(x)-a^2*diff(y(x),x)-2*diff(diff(y(x),x),x)+diff(diff(diff(y(x),x),x),x) = sinh(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (6 c_3 \,a^{2}-6 c_3 \right ) {\mathrm e}^{-a x}+\left (6 c_2 \,a^{2}-6 c_2 \right ) {\mathrm e}^{a x}+\left (6 c_1 \,a^{2}-6 c_1 \right ) {\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-{\mathrm e}^{-x}}{6 a^{2}-6} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to \frac {e^{-x}-3 e^x}{6-6 a^2}+c_1 e^{-a x}+c_3 e^{a x}+c_2 e^{2 x} \end{align*}

✓ Sympy. Time used: 0.408 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{1} e^{2 x} + C_{2} e^{- a x} + C_{3} e^{a x} + \frac {2 \sinh {\left (x \right )}}{3 \left (a^{2} - 1\right )} + \frac {\cosh {\left (x \right )}}{3 \left (a^{2} - 1\right )} \]