75.17.6 problem 556

Internal problem ID [16976]
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 : 556
Date solved : Thursday, March 13, 2025 at 09:04:18 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

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

\[ y = \frac {\left (-8 c_{2} -1\right ) \cos \left (2 x \right )}{16}+\frac {\left (4 c_{1} -x \right ) \sin \left (2 x \right )}{8}+c_{3} +\frac {{\mathrm e}^{2 x}}{16} \]

✓ Mathematica. Time used: 60.164 (sec). Leaf size: 102

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

\[ y(x)\to \int _1^x\left (c_1 \cos (2 K[3])+\int _1^{K[3]}-\frac {1}{2} \sin (2 K[1]) \left (\sin (2 K[1])+e^{2 K[1]}\right )dK[1] \cos (2 K[3])+c_2 \sin (2 K[3])+\sin (2 K[3]) \int _1^{K[3]}\frac {1}{2} \cos (2 K[2]) \left (\sin (2 K[2])+e^{2 K[2]}\right )dK[2]\right )dK[3]+c_3 \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 27

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

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