83.50.3 problem Ex 6 page 150

Internal problem ID [19641]
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 6 page 150
Date solved : Tuesday, January 28, 2025 at 02:05:58 PM
CAS classification : system_of_ODEs

\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+11 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+8 x \left (t \right )+24 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 64

dsolve([4*diff(x(t),t)+9*diff(y(t),t)+11*x(t)+31*y(t)=exp(t),3*diff(x(t),t)+7*diff(y(t),t)+8*x(t)+24*y(t)=exp(2*t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} c_2 +{\mathrm e}^{-4 t} t c_1 +\frac {31 \,{\mathrm e}^{t}}{25}-\frac {49 \,{\mathrm e}^{2 t}}{36} \\ y \left (t \right ) &= \frac {19 \,{\mathrm e}^{2 t}}{36}-\frac {11 \,{\mathrm e}^{t}}{25}-{\mathrm e}^{-4 t} c_2 -{\mathrm e}^{-4 t} t c_1 -{\mathrm e}^{-4 t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.219 (sec). Leaf size: 76

DSolve[{4*D[x[t],t]+9*D[y[t],t]+11*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+8*x[t]+24*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {31 e^t}{25}-\frac {49 e^{2 t}}{36}-e^{-4 t} (c_1 (t-1)+c_2 t) \\ y(t)\to -\frac {11 e^t}{25}+\frac {19 e^{2 t}}{36}+e^{-4 t} ((c_1+c_2) t+c_2) \\ \end{align*}