77.1.163 problem 190 (page 297)

Internal problem ID [17982]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 190 (page 297)
Date solved : Monday, March 31, 2025 at 04:53:07 PM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )-x \left (t \right )-3 y \left (t \right )&=t\\ t \left (\frac {d}{d t}y \left (t \right )\right )-x \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 45

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

\begin{align*} x \left (t \right ) &= \frac {3 t^{4} c_2 -2 t^{3}+3 c_1}{3 t^{2}} \\ y \left (t \right ) &= -\frac {-t^{4} c_2 +t^{3}+3 c_1}{3 t^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 48

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

\begin{align*} x(t)\to c_2 t^2+\frac {c_1}{t^2}-\frac {2 t}{3} \\ y(t)\to -\frac {-c_2 t^4+t^3+3 c_1}{3 t^2} \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 34

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

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