69.17.37 problem 587

Internal problem ID [18373]
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 : 587
Date solved : Thursday, October 02, 2025 at 03:11:05 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} -4 y^{\prime }+y^{\prime \prime \prime }&=x \,{\mathrm e}^{2 x}+\sin \left (x \right )+x^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 44

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

\[ y = \frac {\left (8 x^{2}+64 c_2 -12 x +7\right ) {\mathrm e}^{2 x}}{128}-\frac {x^{3}}{12}-\frac {{\mathrm e}^{-2 x} c_1}{2}-\frac {x}{8}+c_3 +\frac {\cos \left (x \right )}{5} \]

✓ Mathematica. Time used: 14.24 (sec). Leaf size: 109

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

\begin{align*} y(x)&\to \int _1^xe^{-2 K[3]} \left (e^{4 K[3]} c_1+c_2+e^{4 K[3]} \int _1^{K[3]}\frac {1}{4} e^{-2 K[1]} \left (K[1] \left (K[1]+e^{2 K[1]}\right )+\sin (K[1])\right )dK[1]+\int _1^{K[3]}-\frac {1}{4} e^{2 K[2]} \left (K[2] \left (K[2]+e^{2 K[2]}\right )+\sin (K[2])\right )dK[2]\right )dK[3]+c_3 \end{align*}

✓ Sympy. Time used: 0.238 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} + C_{3} e^{- 2 x} - \frac {x^{3}}{12} - \frac {x}{8} + \left (C_{2} + \frac {x^{2}}{16} - \frac {3 x}{32}\right ) e^{2 x} + \frac {\cos {\left (x \right )}}{5} \]