77.42.1 problem 13

Internal problem ID [20792]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IX. Simultaneous equations. Excercise IX (A) at page 154
Problem number : 13
Date solved : Sunday, October 12, 2025 at 05:48:30 AM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )+y \left (t \right )&=0\\ t \left (\frac {d}{d t}y \left (t \right )\right )+x \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 32

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

\begin{align*} x \left (t \right ) &= \frac {c_1 \,t^{2}+c_2}{t} \\ y \left (t \right ) &= -\frac {c_1 \,t^{2}-c_2}{t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 31

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

\begin{align*} x(t)&\to c_1 t+\frac {c_2}{t}\\ y(t)&\to \frac {c_2}{t}-c_1 t \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 17

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

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