69.18.18 problem 607

Internal problem ID [18393]
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 : 607
Date solved : Thursday, October 02, 2025 at 03:11:19 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 )&=0 \\ y^{\prime }\left (0\right )&=2 \\ y^{\prime \prime }\left (0\right )&=4 \\ y^{\prime \prime \prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 9

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-y(x) = 8*exp(x); 
ic:=[y(0) = 0, D(y)(0) = 2, (D@@2)(y)(0) = 4, (D@@3)(y)(0) = 6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2 x \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 103

ode=D[y[x],{x,4}]-y[x]==8*Exp[x]; 
ic={y[0]==0,Derivative[1][y][0] ==2,Derivative[2][y][0] ==4,Derivative[3][y][0]==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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+2 e^x-2 \sin (x)-2 \cos (x) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 8

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): 0, Subs(Derivative(y(x), x), x, 0): 2, Subs(Derivative(y(x), (x, 2)), x, 0): 4, Subs(Derivative(y(x), (x, 3)), x, 0): 6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 x e^{x} \]