57.9.15 problem 2(c)

Internal problem ID [14423]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.3.1 Nonhomogeneous Equations: Undetermined Coefficients. Exercises page 110
Problem number : 2(c)
Date solved : Thursday, October 02, 2025 at 09:37:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+x&=t^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

ode:=diff(diff(x(t),t),t)+x(t) = t^2; 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 20

ode=D[x[t],{t,2}]+x[t]==t^2; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to t^2+c_1 \cos (t)+c_2 \sin (t)-2 \end{align*}

✓ Sympy. Time used: 0.033 (sec). Leaf size: 17

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

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