67.6.2 problem Problem 4(b)

Internal problem ID [14106]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(b)
Date solved : Tuesday, January 28, 2025 at 06:14:32 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= 3 c_{1} {\mathrm e}^{\frac {t}{3}}-\frac {t^{2}}{2}-4 t +c_{2} \\ y &= -\frac {3 c_{1} {\mathrm e}^{\frac {t}{3}}}{2}+2 t -6+\frac {t^{2}}{2}-c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.293 (sec). Leaf size: 208

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

\begin{align*} x(t)\to \left (2 e^{t/3}-1\right ) \int _1^t\frac {1}{3} \left (-3+4 e^{-\frac {K[1]}{3}}\right ) K[1]dK[1]+2 \left (e^{t/3}-1\right ) \int _1^t\left (K[2]-\frac {2}{3} e^{-\frac {K[2]}{3}} K[2]\right )dK[2]+c_1 \left (2 e^{t/3}-1\right )+2 c_2 \left (e^{t/3}-1\right ) \\ y(t)\to -\left (e^{t/3}-1\right ) \int _1^t\frac {1}{3} \left (-3+4 e^{-\frac {K[1]}{3}}\right ) K[1]dK[1]-\left (e^{t/3}-2\right ) \int _1^t\left (K[2]-\frac {2}{3} e^{-\frac {K[2]}{3}} K[2]\right )dK[2]+c_1 \left (-e^{t/3}\right )-c_2 e^{t/3}+c_1+2 c_2 \\ \end{align*}