71.18.5 problem 5 a

Internal problem ID [14579]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 8. Linear Systems of First-Order Differential Equations. Exercises 8.3 page 379
Problem number : 5 a
Date solved : Tuesday, January 28, 2025 at 06:43:46 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 81

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

\begin{align*} \text {y1}(x)\to e^{-2 x} \left (c_1 \cos \left (\sqrt {3} x\right )-\sqrt {3} c_2 \sin \left (\sqrt {3} x\right )\right ) \\ \text {y2}(x)\to \frac {1}{3} e^{-2 x} \left (3 c_2 \cos \left (\sqrt {3} x\right )+\sqrt {3} c_1 \sin \left (\sqrt {3} x\right )\right ) \\ \end{align*}