84.42.7 problem 31.16

Internal problem ID [22395]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 31. Solutions of linear systems with constant coefficients. Supplementary problems
Problem number : 31.16
Date solved : Thursday, October 02, 2025 at 08:38:14 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=6 t \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=0 \\ x_{2} \left (0\right )&=0 \\ x_{3} \left (0\right )&=12 \\ \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = x__3(t), diff(x__3(t),t) = 6*t]; 
ic:=[x__1(0) = 0, x__2(0) = 0, x__3(0) = 12]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {1}{4} t^{4}+6 t^{2} \\ x_{2} \left (t \right ) &= t^{3}+12 t \\ x_{3} \left (t \right ) &= 3 t^{2}+12 \\ \end{align*}

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 37

ode={D[x1[t],t]==x2[t],D[x2[t],t]==x3[t],D[x3[t],t]==6*t}; 
ic={x1[0]==0,x2[0]==0,x3[0]==12}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{4} t^2 \left (t^2+24\right )\\ \text {x2}(t)&\to t \left (t^2+12\right )\\ \text {x3}(t)&\to 3 \left (t^2+4\right ) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(-x3(t) + Derivative(x2(t), t),0),Eq(-6*t + Derivative(x3(t), t),0)] 
ics = {x1(0): 0, x2(0): 0, x3(0): 12} 
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = \frac {t^{4}}{4} + 6 t^{2}, \ x_{2}{\left (t \right )} = t^{3} + 12 t, \ x_{3}{\left (t \right )} = 3 t^{2} + 12\right ] \]