76.6.6 problem 6

Internal problem ID [17461]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 08:27:40 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.155 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 62

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

\begin{align*} x(t)\to \frac {1}{2} e^{-\frac {1}{2} t (t+2)} \left (c_2 e^{t^2}+2 c_1\right ) \\ y(t)\to \frac {1}{2} e^{-\frac {1}{2} t (t+2)} \left (c_2 e^{t^2}-2 c_1\right ) \\ \end{align*}