75.16.29 problem 502

Internal problem ID [16931]
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 : 502
Date solved : Monday, March 31, 2025 at 03:35:46 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

\[ y = \left (c_3 x +c_1 \right ) \cos \left (\sqrt {2}\, x \right )+\left (c_4 x +c_2 \right ) \sin \left (\sqrt {2}\, x \right )+\frac {\sin \left (2 x \right )}{4} \]

✓ Mathematica. Time used: 0.668 (sec). Leaf size: 272

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

\[ y(x)\to \cos \left (\sqrt {2} x\right ) \int _1^x-\frac {1}{8} \sin (2 K[1]) \left (\sqrt {2} \sin \left (\sqrt {2} K[1]\right )-2 \cos \left (\sqrt {2} K[1]\right ) K[1]\right )dK[1]+\frac {1}{32} e^{-i \left (2+\sqrt {2}\right ) x} \left (16 i \left (e^{2 i x}-e^{2 i \left (1+\sqrt {2}\right ) x}\right ) \int _1^x\frac {1}{8} \sin (2 K[2]) \left (\sqrt {2} \cos \left (\sqrt {2} K[2]\right )+2 K[2] \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+4 e^{i \sqrt {2} x} x+4 e^{i \left (4+\sqrt {2}\right ) x} x+e^{2 i \left (1+\sqrt {2}\right ) x} \left (16 (c_1-i c_3)-\left (\sqrt {2}-2-16 c_2+16 i c_4\right ) x\right )+e^{2 i x} \left (\left (-\sqrt {2}+2+16 c_2+16 i c_4\right ) x+16 (c_1+i c_3)\right )\right ) \]

✓ Sympy. Time used: 0.126 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) \sin {\left (\sqrt {2} x \right )} + \left (C_{3} + C_{4} x\right ) \cos {\left (\sqrt {2} x \right )} + \frac {\sin {\left (2 x \right )}}{4} \]