69.17.33 problem 583

Internal problem ID [18369]
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 : 583
Date solved : Thursday, October 02, 2025 at 03:11:01 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+1&=3 \sin \left (2 x \right )+\cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+1 = 3*sin(2*x)+cos(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 3.58 (sec). Leaf size: 52

ode=D[y[x],{x,2}]+2*D[y[x],x]+1==3*Sin[2*x]+Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.157 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*sin(2*x) - cos(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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