60.4.12 problem 1465

Internal problem ID [11435]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1465
Date solved : Sunday, March 30, 2025 at 08:22:10 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }-a^{2} y^{\prime }+2 a^{2} y-\sinh \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 71

ode:=diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-a^2*diff(y(x),x)+2*a^2*y(x)-sinh(x) = 0; 
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: 0.064 (sec). Leaf size: 192

ode=-Sinh[x] + 2*a^2*y[x] - a^2*D[y[x],x] - 2*D[y[x],{x,2}] + Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-((a+1) x)} \left (6 \left (a^2-4\right ) e^x \int _1^x\frac {e^{a K[1]} \sinh (K[1])}{2 a^2+4 a}dK[1]+6 \left (a^2-4\right ) e^{2 a x+x} \int _1^x\frac {e^{-a K[2]} \sinh (K[2])}{2 a^2-4 a}dK[2]+6 a^2 c_1 e^x+6 a^2 c_2 e^{(a+3) x}+6 a^2 c_3 e^{2 a x+x}-e^{a x}+3 e^{(a+2) x}-24 c_2 e^{(a+3) x}-24 c_3 e^{2 a x+x}-24 c_1 e^x\right )}{6 \left (a^2-4\right )} \]

✓ Sympy. Time used: 0.697 (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)
 

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