78.2.39 problem 16.a

Internal problem ID [20991]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 16.a
Date solved : Thursday, October 02, 2025 at 07:01:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+x&=5 t^{2} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ x^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 15

ode:=diff(diff(x(t),t),t)+x(t) = 5*t^2; 
ic:=[x(0) = 4, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -10+14 \cos \left (t \right )+5 t^{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 17

ode=D[x[t],{t,2}]+x[t]==5*t^2; 
ic={x[0]==4,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 5 \left (t^2-2\right )+14 \cos (t) \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = 5 t^{2} + 14 \cos {\left (t \right )} - 10 \]