60.9.54 problem 1910

Internal problem ID [11833]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1910
Date solved : Sunday, March 30, 2025 at 09:18:12 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.127 (sec). Leaf size: 36

ode:=[t*diff(x(t),t) = 2*x(t)-t, t^3*diff(y(t),t) = -x(t)+t^2*y(t)+t, t^4*diff(z(t),t) = -x(t)-t^2*y(t)+t^3*z(t)+t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_3 \,t^{2}+t \\ y \left (t \right ) &= t c_2 +c_3 \\ z \left (t \right ) &= \frac {c_1 \,t^{2}+t c_2 +c_3}{t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 39

ode={t*D[x[t],t]==2*x[t]-t,t^3*D[y[t],t]==-x[t]+t^2*y[t]+t,t^4*D[z[t],t]==-x[t]-t^2*y[t]+t^3*z[t]+t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to t (1+c_3 t) \\ y(t)\to c_2 t+c_3 \\ z(t)\to c_1 t+\frac {c_3}{t}+c_2 \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(t*Derivative(x(t), t) + t - 2*x(t),0),Eq(t**3*Derivative(y(t), t) - t**2*y(t) - t + x(t),0),Eq(t**4*Derivative(z(t), t) - t**3*z(t) + t**2*y(t) - t + x(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

ValueError : substitution cannot create dummy dependencies