83.15.17 problem 6 (b)

Internal problem ID [22041]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 6 (b)
Date solved : Sunday, October 12, 2025 at 05:52:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=1\\ \frac {d}{d t}x \left (t \right )+x \left (t \right )+\frac {d^{2}}{d t^{2}}y \left (t \right )-9 y \left (t \right )+\frac {d}{d t}z \left (t \right )+z \left (t \right )&=0\\ 5 x \left (t \right )+\frac {d^{2}}{d t^{2}}z \left (t \right )-4 z \left (t \right )&=2 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 68

ode:=[diff(diff(x(t),t),t) = 1, diff(x(t),t)+x(t)+diff(diff(y(t),t),t)-9*y(t)+diff(z(t),t)+z(t) = 0, 5*x(t)+diff(diff(z(t),t),t)-4*z(t) = 2]; 
ic:=[x(0) = 0, y(0) = 0, z(0) = 0, D(x)(0) = 0, D(y)(0) = 0, D(z)(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{2} \\ y \left (t \right ) &= \frac {9 \,{\mathrm e}^{2 t}}{160}+\frac {1}{144}+\frac {t^{2}}{8}+\frac {t}{4}-\frac {3 \,{\mathrm e}^{-2 t}}{160}-\frac {4 \,{\mathrm e}^{3 t}}{45}+\frac {2 \,{\mathrm e}^{-3 t}}{45} \\ z \left (t \right ) &= \frac {3 \,{\mathrm e}^{-2 t}}{32}+\frac {3 \,{\mathrm e}^{2 t}}{32}-\frac {3}{16}+\frac {5 t^{2}}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 67

ode={D[x[t],{t,2}]==0,D[x[t],t]+x[t]+D[y[t],{t,2}]-9*y[t]+D[z[t],t]+z[t]==0,5*x[t]+D[z[t],{t,2}]-4*z[t]==2}; 
ic={x[0]==0,y[0]==0,z[0]==0,Derivative[1][x][0] ==0,Derivative[1][y][0] ==0,Derivative[1][z][0] ==0}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 0\\ y(t)&\to -\frac {1}{180} e^{-3 t} \left (e^t-1\right )^3 \left (15 e^t+21 e^{2 t}+16 e^{3 t}+8\right )\\ z(t)&\to \frac {1}{4} e^{-2 t} \left (e^{2 t}-1\right )^2 \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(Derivative(x(t), (t, 2)) - 1,0),Eq(x(t) - 9*y(t) + z(t) + Derivative(x(t), t) + Derivative(y(t), (t, 2)) + Derivative(z(t), t),0),Eq(5*x(t) - 4*z(t) + Derivative(z(t), (t, 2)) - 2,0)] 
ics = {y(0): 0, z(0): 0, Subs(Derivative(x(t), t), t, 0): 1, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(z(t), t), t, 0): 1} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

NotAlgebraic : Integral(5/4, (t, 0)) does not seem to be an algebraic element