71.17.6 problem 13 (a)

Internal problem ID [14562]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 7. Systems of First-Order Differential Equations. Exercises page 329
Problem number : 13 (a)
Date solved : Tuesday, January 28, 2025 at 06:43:29 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.387 (sec). Leaf size: 118

dsolve([diff(y__1(x),x) = 3*y__1(x)-2*y__2(x), diff(y__2(x),x) = -y__1(x)+y__2(x), y__1(0) = 1, y__2(0) = -1], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 79

DSolve[{D[ y1[x],x]==3*y1[x]-2*y2[x],D[ y2[x],x]==-y1[x]+y2[x]},{y1[0]==1,y2[0]==-1},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (\left (1+\sqrt {3}\right ) e^{2 \sqrt {3} x}+1-\sqrt {3}\right ) \\ \text {y2}(x)\to -\frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (e^{2 \sqrt {3} x}+1\right ) \\ \end{align*}