60.9.55 problem 1910

Internal problem ID [11909]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1910
Date solved : Tuesday, January 28, 2025 at 06:24:04 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*}

✓ Solution by Maple

Time used: 0.132 (sec). Leaf size: 36

dsolve([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],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_3 \,t^{2}+t \\ y \left (t \right ) &= c_{2} t +c_3 \\ z &= \frac {c_{1} t^{2}+c_{2} t +c_3}{t} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{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},{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*}