82.28.2 problem Ex. 3

Internal problem ID [18810]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 75
Problem number : Ex. 3
Date solved : Monday, March 31, 2025 at 06:15:29 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y&=\left ({\mathrm e}^{x}+1\right )^{2} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 55

ode:=diff(diff(diff(y(x),x),x),x)-y(x) = (exp(x)+1)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{2 x}}{7}+\frac {4 \ln \left ({\mathrm e}^{\frac {x}{2}}\right ) {\mathrm e}^{x}}{3}-1-\frac {2 \,{\mathrm e}^{x}}{3}+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \]

✓ Mathematica. Time used: 0.413 (sec). Leaf size: 79

ode=D[y[x],{x,3}]-y[x]==(Exp[x]+1)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {2 e^x x}{3}-\frac {2 e^x}{3}+\frac {e^{2 x}}{7}+c_1 e^x+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )-1 \]

✓ Sympy. Time used: 0.162 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(exp(x) + 1)**2 - y(x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + \frac {2 x}{3}\right ) e^{x} + \left (C_{2} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{3} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \frac {e^{2 x}}{7} - 1 \]