75.18.9 problem 598

Internal problem ID [17026]
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. Initial value problem. Exercises page 140
Problem number : 598
Date solved : Monday, March 31, 2025 at 03:38:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+4*y(x) = sin(x); 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 76

ode=D[y[x],{x,2}]+4*y[x]==Sin[x]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\sin (2 x) \int _1^0\frac {1}{2} \cos (2 K[1]) \sin (K[1])dK[1]+\sin (2 x) \int _1^x\frac {1}{2} \cos (2 K[1]) \sin (K[1])dK[1]+\cos (2 x)-\frac {1}{3} \sin ^3(x) \cos (2 x)+\sin (x) \cos (x) \]

✓ Sympy. Time used: 0.094 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - sin(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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