67.5.13 problem Problem 3(b)

Internal problem ID [14099]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 3(b)
Date solved : Tuesday, January 28, 2025 at 06:14:26 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y}{4}\\ y^{\prime }&=\frac {x \left (t \right )}{2}-\frac {3 y}{2} \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 85

dsolve([diff(x(t),t)=5/4*x(t)+3/4*y(t),diff(y(t),t)=1/2*x(t)-3/2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \\ y &= \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}} \sqrt {145}}{6}-\frac {11 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {145}\right ) t}{8}}}{6}-\frac {11 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {145}\right ) t}{8}}}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 161

DSolve[{D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==1/2*x[t]-3/2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{290} e^{-\frac {1}{8} \left (1+\sqrt {145}\right ) t} \left (c_1 \left (\left (145+11 \sqrt {145}\right ) e^{\frac {\sqrt {145} t}{4}}+145-11 \sqrt {145}\right )+6 \sqrt {145} c_2 \left (e^{\frac {\sqrt {145} t}{4}}-1\right )\right ) \\ y(t)\to \frac {1}{290} e^{-\frac {1}{8} \left (1+\sqrt {145}\right ) t} \left (4 \sqrt {145} c_1 \left (e^{\frac {\sqrt {145} t}{4}}-1\right )-c_2 \left (\left (11 \sqrt {145}-145\right ) e^{\frac {\sqrt {145} t}{4}}-145-11 \sqrt {145}\right )\right ) \\ \end{align*}