75.27.10 problem 785

Internal problem ID [17246]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 785
Date solved : Tuesday, January 28, 2025 at 08:27:36 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 46

dsolve([diff(x(t),t$2)=3*x(t)+y(t),diff(y(t),t)=-2*x(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 125

DSolve[{D[x[t],{t,2}]==3*x[t]+y[t],D[y[t],t]==-2*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (e^{3 t} (3 t+5)+4\right )+c_2 \left (e^{3 t} (3 t+2)-2\right )+c_3 \left (e^{3 t} (3 t-1)+1\right )\right ) \\ y(t)\to \frac {1}{9} e^{-2 t} \left (c_1 \left (4-2 e^{3 t} (3 t+2)\right )+c_2 \left (e^{3 t} (2-6 t)-2\right )+c_3 \left (e^{3 t} (8-6 t)+1\right )\right ) \\ \end{align*}