70.15.23 problem 24

Internal problem ID [18951]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 24
Date solved : Thursday, October 02, 2025 at 03:34:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+y(t) = t*(1+sin(t)); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.231 (sec). Leaf size: 59

ode=D[y[t],{t,2}]+y[t]==t*(1+Sin[t]); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (t) \int _1^t-K[1] \sin (K[1]) (\sin (K[1])+1)dK[1]+\sin (t) \int _1^t\cos (K[2]) K[2] (\sin (K[2])+1)dK[2]+c_1 \cos (t)+c_2 \sin (t) \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 22

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

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