90.33.2 problem 2

Internal problem ID [25476]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 645
Problem number : 2
Date solved : Friday, October 03, 2025 at 12:01:56 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{2} \left (t \right )+t^{2}\\ y_{2}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right )+1 \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 49

ode:=[diff(y__1(t),t) = y__1(t)+y__2(t)+t^2, diff(y__2(t),t) = -y__1(t)+y__2(t)+1]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{t} \sin \left (t \right ) c_2 +{\mathrm e}^{t} \cos \left (t \right ) c_1 -\frac {t^{2}}{2}+1 \\ y_{2} \left (t \right ) &= -\frac {t^{2}}{2}+{\mathrm e}^{t} \cos \left (t \right ) c_2 -{\mathrm e}^{t} \sin \left (t \right ) c_1 -t -1 \\ \end{align*}

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 62

ode={D[y1[t],t]==y1[t]+y2[t]+t^2,D[y2[t],t]==-y1[t]+y2[t]+1}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -\frac {t^2}{2}+c_1 e^t \cos (t)+c_2 e^t \sin (t)+1\\ \text {y2}(t)&\to -\frac {t^2}{2}-t+c_2 e^t \cos (t)-c_1 e^t \sin (t)-1 \end{align*}

✓ Sympy. Time used: 0.324 (sec). Leaf size: 112

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-t**2 - y1(t) - y2(t) + Derivative(y1(t), t),0),Eq(y1(t) - y2(t) + Derivative(y2(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = C_{1} e^{t} \sin {\left (t \right )} + C_{2} e^{t} \cos {\left (t \right )} - \frac {t^{2} \sin ^{2}{\left (t \right )}}{2} - \frac {t^{2} \cos ^{2}{\left (t \right )}}{2} + \sin ^{2}{\left (t \right )} + \cos ^{2}{\left (t \right )}, \ y_{2}{\left (t \right )} = C_{1} e^{t} \cos {\left (t \right )} - C_{2} e^{t} \sin {\left (t \right )} - \frac {t^{2} \sin ^{2}{\left (t \right )}}{2} - \frac {t^{2} \cos ^{2}{\left (t \right )}}{2} - t \sin ^{2}{\left (t \right )} - t \cos ^{2}{\left (t \right )} - \sin ^{2}{\left (t \right )} - \cos ^{2}{\left (t \right )}\right ] \]