85.1.17 problem 11 (c)

Internal problem ID [22422]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. A Exercises at page 12
Problem number : 11 (c)
Date solved : Thursday, October 02, 2025 at 08:39:10 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=t^{2}-4 t +8 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ x^{\prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 24

ode:=diff(diff(x(t),t),t) = t^2-4*t+8; 
ic:=[x(0) = 1, D(x)(0) = -3]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {1}{12} t^{4}-\frac {2}{3} t^{3}+4 t^{2}-3 t +1 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 29

ode=D[x[t],{t,2}]==8-4*t+t^2; 
ic={x[0]==1,Derivative[1][x][0] ==-3}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {t^4}{12}-\frac {2 t^3}{3}+4 t^2-3 t+1 \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**2 + 4*t + Derivative(x(t), (t, 2)) - 8,0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): -3} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {t^{4}}{12} - \frac {2 t^{3}}{3} + 4 t^{2} - 3 t + 1 \]