23.4.7 problem 8(g)

Internal problem ID [4172]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(g)
Date solved : Monday, January 27, 2025 at 08:41:16 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 85

dsolve([diff(y__1(x),x)=y__2(x)-y__1(x),diff(y__2(x),x)=3*y__1(x)-4*y__2(x)],singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) x}{2}} \\ y_{2} \left (x \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) x}{2}} \sqrt {21}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) x}{2}} \sqrt {21}}{2}-\frac {3 c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) x}{2}}}{2}-\frac {3 c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) x}{2}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 146

DSolve[{D[y1[x],x]==y2[x]-y1[x],D[y2[x],x]==3*y1[x]-4*y2[x]},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{42} e^{-\frac {1}{2} \left (5+\sqrt {21}\right ) x} \left (3 c_1 \left (\left (7+\sqrt {21}\right ) e^{\sqrt {21} x}+7-\sqrt {21}\right )+2 \sqrt {21} c_2 \left (e^{\sqrt {21} x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{14} e^{-\frac {1}{2} \left (5+\sqrt {21}\right ) x} \left (2 \sqrt {21} c_1 \left (e^{\sqrt {21} x}-1\right )-c_2 \left (\left (\sqrt {21}-7\right ) e^{\sqrt {21} x}-7-\sqrt {21}\right )\right ) \\ \end{align*}