60.9.23 problem 1878

Internal problem ID [11877]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1878
Date solved : Tuesday, January 28, 2025 at 06:23:57 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 \left (\frac {d}{d t}y \left (t \right )\right )-\left (t +2\right ) x \left (t \right )-y \left (t \right ) t&=-t \end{align*}

✓ Solution by Maple

Time used: 0.087 (sec). Leaf size: 38

dsolve([t*diff(x(t),t)+2*x(t)=t,t*diff(y(t),t)-(t+2)*x(t)-t*y(t)=-t],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {t}{3}+\frac {c_{2}}{t^{2}} \\ y \left (t \right ) &= \frac {3 c_{1} {\mathrm e}^{t} t^{2}-t^{3}-3 c_{2}}{3 t^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 39

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

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