60.9.24 problem 1879

Internal problem ID [11803]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1879
Date solved : Sunday, March 30, 2025 at 09:15:43 PM
CAS classification : system_of_ODEs

\begin{align*} t \left (\frac {d}{d t}x \left (t \right )\right )+2 x \left (t \right )-2 y \left (t \right )&=t\\ t \left (\frac {d}{d t}y \left (t \right )\right )+x \left (t \right )+5 y \left (t \right )&=t^{2} \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 53

ode:=[t*diff(x(t),t)+2*x(t)-2*y(t) = t, t*diff(y(t),t)+x(t)+5*y(t) = t^2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {2 t^{6}+9 t^{5}+30 c_1 t +30 c_2}{30 t^{4}} \\ y \left (t \right ) &= -\frac {-8 t^{6}+3 t^{5}+30 c_1 t +60 c_2}{60 t^{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 58

ode={t*D[x[t],t]+2*(x[t]-y[t])==t,t*D[y[t],t]+x[t]+5*y[t]==t^2}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {c_1}{t^4}+\frac {c_2}{t^3}+\frac {1}{30} t (2 t+9) \\ y(t)\to -\frac {-8 t^6+3 t^5+30 c_2 t+60 c_1}{60 t^4} \\ \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(t*Derivative(x(t), t) - t + 2*x(t) - 2*y(t),0),Eq(-t**2 + t*Derivative(y(t), t) + x(t) + 5*y(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1}}{t^{4}} - \frac {2 C_{2}}{t^{3}} + \frac {t^{2}}{15} + \frac {3 t}{10}, \ y{\left (t \right )} = \frac {C_{1}}{t^{4}} + \frac {C_{2}}{t^{3}} + \frac {2 t^{2}}{15} - \frac {t}{20}\right ] \]