83.50.4 problem Ex 7 page 150

Internal problem ID [19642]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IX. Simultaneous equations
Problem number : Ex 7 page 150
Date solved : Tuesday, January 28, 2025 at 08:33:06 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 38

dsolve([t*diff(x(t),t)=t-2*x(t),t*diff(y(t),t)=t*x(t)+t*y(t)+2*x(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: 0.056 (sec). Leaf size: 39

DSolve[{t*D[x[t],t]==t-2*x[t],t*D[y[t],t]==t*x[t]+t*y[t]+2*x[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*}