84.41.4 problem 31.5

Internal problem ID [22386]
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. Solved problems. Page 194
Problem number : 31.5
Date solved : Thursday, October 02, 2025 at 08:38:07 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=-9 x \left (t \right )+6 y+t \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 41

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -9*x(t)+6*y(t)+t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} c_2 +{\mathrm e}^{3 t} t c_1 +\frac {t}{9}+\frac {2}{27} \\ y \left (t \right ) &= \frac {1}{9}+{\mathrm e}^{3 t} \left (3 t c_1 +c_1 +3 c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 62

ode={D[x[t],t]==y[t],D[y[t],t]==-9*x[t]+6*y[t]+t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {t}{9}+(c_2-3 c_1) e^{3 t} t+c_1 e^{3 t}+\frac {2}{27}\\ y(t)&\to \frac {1}{9}+e^{3 t} (-9 c_1 t+3 c_2 t+c_2) \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(-t + 9*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{1} t e^{3 t} + \frac {t}{9} + \left (C_{1} - 3 C_{2}\right ) e^{3 t} + \frac {2}{27}, \ y{\left (t \right )} = - 9 C_{1} t e^{3 t} - 9 C_{2} e^{3 t} + \frac {1}{9}\right ] \]