14.20.2 problem 2

Internal problem ID [2699]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 2
Date solved : Monday, January 27, 2025 at 06:11:59 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y+t\\ y^{\prime }&=-4 x \left (t \right )+3 y-1 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 41

dsolve([diff(x(t),t)=-2*x(t)+y(t)+t,diff(y(t),t)=-4*x(t)+3*y(t)-1],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{2 t}+{\mathrm e}^{-t} c_1 +\frac {3 t}{2}-\frac {3}{4} \\ y &= 4 c_2 \,{\mathrm e}^{2 t}+{\mathrm e}^{-t} c_1 +2 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.081 (sec). Leaf size: 190

DSolve[{D[x[t],t]==-2*x[t]-y[t]+t,D[y[t],t]==-4*x[t]+3*y[t]-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {3 t}{10}+\frac {1}{82} \left (\left (41-5 \sqrt {41}\right ) c_1-2 \sqrt {41} c_2\right ) e^{\frac {1}{2} \left (1+\sqrt {41}\right ) t}+\frac {1}{82} \left (\left (41+5 \sqrt {41}\right ) c_1+2 \sqrt {41} c_2\right ) e^{\frac {1}{2} \left (t-\sqrt {41} t\right )}-\frac {23}{100} \\ y(t)\to \frac {2 t}{5}+\frac {1}{82} \left (8 \sqrt {41} c_1+\left (41-5 \sqrt {41}\right ) c_2\right ) e^{\frac {1}{2} \left (t-\sqrt {41} t\right )}+\frac {1}{82} \left (\left (41+5 \sqrt {41}\right ) c_2-8 \sqrt {41} c_1\right ) e^{\frac {1}{2} \left (1+\sqrt {41}\right ) t}+\frac {4}{25} \\ \end{align*}