80.7.33 problem D 2

Internal problem ID [21352]
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 2
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 )+y \left (t \right )&=3 t\\ \frac {d}{d t}y \left (t \right )-t \left (\frac {d}{d t}x \left (t \right )\right )&=0 \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 75

ode:=[diff(x(t),t)+y(t) = 3*t, diff(y(t),t)-t*diff(x(t),t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 125

ode={D[x[t],t]+y[t]==3*t,D[y[t],t]-t*D[x[t],t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {3}{2} t^2 \, _2F_2\left (1,1;\frac {1}{2},2;\frac {t^2}{2}\right )-\frac {1}{2} \text {erf}\left (\frac {t}{\sqrt {2}}\right ) \left (-3 \pi \text {erfi}\left (\frac {t}{\sqrt {2}}\right )+\sqrt {2 \pi } \left (3 e^{\frac {t^2}{2}} t+c_1\right )\right )+c_2\\ y(t)&\to -3 \sqrt {\frac {\pi }{2}} e^{-\frac {t^2}{2}} \text {erfi}\left (\frac {t}{\sqrt {2}}\right )+c_1 e^{-\frac {t^2}{2}}+3 t \end{align*}

✓ Sympy. Time used: 46.858 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*t + y(t) + Derivative(x(t), t),0),Eq(-t*Derivative(x(t), t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} - \frac {\sqrt {2} \sqrt {\pi } C_{2} \operatorname {erf}{\left (\frac {\sqrt {2} t}{2} \right )}}{2} + \frac {3 t^{2} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ \frac {3}{2}, 2 \end {matrix}\middle | {- \frac {t^{2}}{2}} \right )}}{2}, \ y{\left (t \right )} = C_{2} e^{- \frac {t^{2}}{2}} + 3 t - \frac {3 \sqrt {2} \sqrt {\pi } e^{- \frac {t^{2}}{2}} \operatorname {erfi}{\left (\frac {\sqrt {2} t}{2} \right )}}{2}\right ] \]