75.16.13 problem 486

Internal problem ID [16915]
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 : 486
Date solved : Monday, March 31, 2025 at 03:35:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

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

\[ y = -\frac {\left (\left (x -4 c_1 +\frac {1}{2}\right ) \cos \left (2 x \right )+\sin \left (2 x \right ) \left (x -4 c_2 \right )\right ) {\mathrm e}^{2 x}}{4} \]

✓ Mathematica. Time used: 0.952 (sec). Leaf size: 94

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

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

✓ Sympy. Time used: 0.397 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (2 x \right )} + C_{2} \cos {\left (2 x \right )} - \frac {\sqrt {2} x \sin {\left (2 x + \frac {\pi }{4} \right )}}{4}\right ) e^{2 x} \]