69.16.26 problem 499

Internal problem ID [18286]
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. Trial and error method. Exercises page 132
Problem number : 499
Date solved : Thursday, October 02, 2025 at 03:10:07 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime }&={\mathrm e}^{4 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x) = exp(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \left (2+x \right )+c_2 \right ) {\mathrm e}^{-x}+c_3 x +c_4 +\frac {{\mathrm e}^{4 x}}{400} \]

✓ Mathematica. Time used: 60.055 (sec). Leaf size: 76

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

\begin{align*} y(x)&\to \int _1^x\int _1^{K[3]}\frac {1}{5} e^{-K[2]} \left (5 c_1+\left (5 c_2+e^{5 K[2]}\right ) K[2]+5 \int _1^{K[2]}-e^{5 K[1]} K[1]dK[1]\right )dK[2]dK[3]+c_4 x+c_3 \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} + C_{4} e^{- x} + x \left (C_{2} + C_{3} e^{- x}\right ) + \frac {e^{4 x}}{400} \]