75.16.55 problem 528

Internal problem ID [16949]
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 : 528
Date solved : Thursday, March 13, 2025 at 09:01:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }-2 y&=8 \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 38

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)-2*y(x) = 8*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\left (-2+\sqrt {6}\right ) x} c_{2} +{\mathrm e}^{-\left (2+\sqrt {6}\right ) x} c_{1} -\frac {16 \cos \left (2 x \right )}{25}-\frac {12 \sin \left (2 x \right )}{25} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 52

ode=D[y[x],{x,2}]+4*D[y[x],x]-2*y[x]==8*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 e^{-\left (\left (2+\sqrt {6}\right ) x\right )}+c_2 e^{\left (\sqrt {6}-2\right ) x}-\frac {4}{25} (3 \sin (2 x)+4 \cos (2 x)) \]

✓ Sympy. Time used: 0.228 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) - 8*sin(2*x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x \left (-2 + \sqrt {6}\right )} + C_{2} e^{- x \left (2 + \sqrt {6}\right )} - \frac {12 \sin {\left (2 x \right )}}{25} - \frac {16 \cos {\left (2 x \right )}}{25} \]