83.15.13 problem 5 (a)

Internal problem ID [22037]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 5 (a)
Date solved : Thursday, October 02, 2025 at 08:21:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&={\frac {7}{9}} \\ y \left (\frac {\pi }{2}\right )&=\frac {\pi }{6}-1 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)+4*y(x) = x*sin(x); 
ic:=[y(0) = 7/9, y(1/2*Pi) = 1/6*Pi-1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sin \left (2 x \right ) c_2 +\cos \left (2 x \right )-\frac {2 \cos \left (x \right )}{9}+\frac {x \sin \left (x \right )}{3} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 30

ode=D[y[x],{x,2}]+4*y[x]==x*Sin[x]; 
ic={y[0]==7/9,y[Pi/2]==Pi/6-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} x \sin (x)+\cos (2 x)+\cos (x) \left (-\frac {2}{9}+2 c_2 \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sin(x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 7/9, y(pi/2): -1 + pi/6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (2 x \right )} + \frac {x \sin {\left (x \right )}}{3} - \frac {2 \cos {\left (x \right )}}{9} + \cos {\left (2 x \right )} \]