86.12.2 problem 2

Internal problem ID [23209]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 10. The Laplace transform. Exercise 10c at page 156
Problem number : 2
Date solved : Sunday, October 12, 2025 at 05:55:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d^{2}}{d t^{2}}y \left (t \right )&=t\\ \frac {d^{2}}{d t^{2}}x \left (t \right )-\frac {d^{2}}{d t^{2}}y \left (t \right )&=3 t \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ D\left (x \right )\left (0\right )&=0 \\ y \left (0\right )&=0 \\ D\left (y \right )\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 17

ode:=[diff(diff(x(t),t),t)+diff(diff(y(t),t),t) = t, diff(diff(x(t),t),t)-diff(diff(y(t),t),t) = 3*t]; 
ic:=[x(0) = 0, D(x)(0) = 0, y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {t^{3}}{3} \\ y \left (t \right ) &= -\frac {t^{3}}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 22

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

\begin{align*} x(t)&\to \frac {t^3}{3}\\ y(t)&\to -\frac {t^3}{6} \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 17

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

\[ \left [ x{\left (t \right )} = \frac {t^{3}}{12}, \ y{\left (t \right )} = - \frac {5 t^{3}}{12}\right ] \]