84.40.6 problem 27.12

Internal problem ID [22381]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 27. Solutions of systems of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 27.12
Date solved : Sunday, October 12, 2025 at 05:52:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}w \left (t \right )-2 z \left (t \right )&=0\\ \frac {d}{d t}w \left (t \right )+\frac {d}{d t}y \left (t \right )-z \left (t \right )&=2 t\\ \frac {d}{d t}w \left (t \right )-2 y \left (t \right )+\frac {d^{2}}{d t^{2}}z \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} w \left (0\right )&=0 \\ D\left (w \right )\left (0\right )&=0 \\ z \left (0\right )&=1 \\ D\left (z \right )\left (0\right )&=0 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 15

ode:=[diff(diff(w(t),t),t)-2*z(t) = 0, diff(w(t),t)+diff(y(t),t)-z(t) = 2*t, diff(w(t),t)-2*y(t)+diff(diff(z(t),t),t) = 0]; 
ic:=[w(0) = 0, D(w)(0) = 0, z(0) = 1, D(z)(0) = 0, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} w \left (t \right ) &= t^{2} \\ y \left (t \right ) &= t \\ z \left (t \right ) &= 1 \\ \end{align*}

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 18

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

\begin{align*} w(t)&\to t^2+2\\ y(t)&\to t\\ z(t)&\to 1 \end{align*}

✓ Sympy. Time used: 0.880 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
w = Function("w") 
z = Function("z") 
y = Function("y") 
ode=[Eq(-2*z(t) + Derivative(w(t), (t, 2)),0),Eq(-2*t - z(t) + Derivative(w(t), t) + Derivative(y(t), t),0),Eq(-2*y(t) + Derivative(w(t), t) + Derivative(z(t), (t, 2)),0)] 
ics = {w(0): 2, Subs(Derivative(w(t), t), t, 0): 0, z(0): 1, Subs(Derivative(z(t), t), t, 0): 0, y(0): 0} 
dsolve(ode,func=[w(t),z(t),y(t)],ics=ics)
 

\[ \left [ w{\left (t \right )} = t^{2} + 2, \ z{\left (t \right )} = \sin ^{2}{\left (t \right )} + \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )}\right ] \]