40.15.1 problem 10

Internal problem ID [6787]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 21. System of simultaneous linear equations. Supplemetary problems. Page 163
Problem number : 10
Date solved : Monday, January 27, 2025 at 02:30:32 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.083 (sec). Leaf size: 72

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

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