80.7.32 problem D 1

Internal problem ID [21351]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : D 1
Date solved : Sunday, October 12, 2025 at 05:51:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+t y \left (t \right )&=-1\\ \frac {d}{d t}y \left (t \right )+\frac {d}{d t}x \left (t \right )&=2 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 69

ode:=[diff(x(t),t)+t*y(t) = -1, diff(y(t),t)+diff(x(t),t) = 2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -\frac {3 \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right ) {\mathrm e}^{\frac {t^{2}}{2}}}{2}-c_2 \,{\mathrm e}^{\frac {t^{2}}{2}}+2 t +c_1 \\ y \left (t \right ) &= \frac {\left (3 \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )+2 c_2 \right ) {\mathrm e}^{\frac {t^{2}}{2}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 90

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

\begin{align*} x(t)&\to -3 \sqrt {\frac {\pi }{2}} e^{\frac {t^2}{2}} \text {erf}\left (\frac {t}{\sqrt {2}}\right )-c_1 e^{\frac {t^2}{2}}+2 t+c_2\\ y(t)&\to \frac {1}{2} e^{\frac {t^2}{2}} \left (3 \sqrt {2 \pi } \text {erf}\left (\frac {t}{\sqrt {2}}\right )+2 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.192 (sec). Leaf size: 88

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

\[ \left [ x{\left (t \right )} = C_{1} - C_{2} e^{\frac {t^{2}}{2}} + C_{2} + 2 t - \frac {3 \sqrt {2} \sqrt {\pi } e^{\frac {t^{2}}{2}} \operatorname {erf}{\left (\frac {\sqrt {2} t}{2} \right )}}{2}, \ y{\left (t \right )} = C_{2} e^{\frac {t^{2}}{2}} + \frac {3 \sqrt {2} \sqrt {\pi } e^{\frac {t^{2}}{2}} \operatorname {erf}{\left (\frac {\sqrt {2} t}{2} \right )}}{2}\right ] \]