83.15.12 problem 4 (L)

Internal problem ID [22036]
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 : 4 (L)
Date solved : Thursday, October 02, 2025 at 08:21:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {7}{9}} \\ y^{\prime }\left (0\right )&=-{\frac {5}{2}} \\ \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)+4*y(t) = t*sin(t); 
ic:=[y(0) = 7/9, D(y)(0) = -5/2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \cos \left (2 t \right )-\frac {5 \sin \left (2 t \right )}{4}-\frac {2 \cos \left (t \right )}{9}+\frac {t \sin \left (t \right )}{3} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 29

ode=D[y[t],{t,2}]+4*y[t]==t*Sin[t]; 
ic={y[0]==7/9,Derivative[1][y][0] ==-5/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{3} t \sin (t)+\cos (2 t)-\frac {1}{18} (45 \sin (t)+4) \cos (t) \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*sin(t) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 7/9, Subs(Derivative(y(t), t), t, 0): -5/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t \sin {\left (t \right )}}{3} - \frac {5 \sin {\left (2 t \right )}}{4} - \frac {2 \cos {\left (t \right )}}{9} + \cos {\left (2 t \right )} \]