75.17.5 problem 555

Internal problem ID [16975]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Superposition principle. Exercises page 137
Problem number : 555
Date solved : Thursday, March 13, 2025 at 09:04:18 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }&=1+{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

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

\[ y = \left (x -2+c_{1} \right ) {\mathrm e}^{x}-\frac {x^{2}}{2}+c_{2} x +c_{3} \]

✓ Mathematica. Time used: 1.547 (sec). Leaf size: 69

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

\begin{align*} y(x)\to \int _1^x\int _1^{K[2]}\left (e^{K[1]} (c_1+K[1])-1\right )dK[1]dK[2]+c_3 x+c_2 \\ y(x)\to -\frac {x^2}{2}+x+e^x (x-2)+c_3 x+e-\frac {1}{2}+c_2 \\ \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 20

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

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