75.18.16 problem 605

Internal problem ID [17033]
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. Initial value problem. Exercises page 140
Problem number : 605
Date solved : Monday, March 31, 2025 at 03:38:44 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-y&=8 \,{\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=1\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 23

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-y(x) = 8*exp(x); 
ic:=y(0) = -1, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x}+\left (2 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right )+2 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 104

ode=D[y[x],{x,4}]-y[x]==8*Exp[x]; 
ic={y[0]==-1,Derivative[1][y][0] ==0,Derivative[2][y][0] ==1,Derivative[3][y][0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\sin (x) \int _1^0-4 e^{K[2]} \cos (K[2])dK[2]+\sin (x) \int _1^x-4 e^{K[2]} \cos (K[2])dK[2]-\cos (x) \int _1^04 e^{K[1]} \sin (K[1])dK[1]+\cos (x) \int _1^x4 e^{K[1]} \sin (K[1])dK[1]+2 e^x x+e^{-x}-e^x-\cos (x) \]

✓ Sympy. Time used: 0.161 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) - 8*exp(x) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 1, Subs(Derivative(y(x), (x, 3)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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