71.18.9 problem 7

Internal problem ID [14583]
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 : 7
Date solved : Tuesday, January 28, 2025 at 06:43:48 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 64

dsolve([diff(y__1(x),x)=2*y__1(x)+y__2(x)-2*y__3(x),diff(y__2(x),x)=3*y__2(x)-2*y__3(x),diff(y__3(x),x)=3*y__1(x)+y__2(x)-3*y__3(x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 159

DSolve[{D[ y1[x],x]==2*y1[x]+y2[x]-2*y3[x],D[ y2[x],x]==3*y2[x]-2*y3[x],D[ y3[x],x]==3*y1[x]+y2[x]-3*y3[x]},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to e^{-x} \left (\left (e^x-1\right ) \left (c_2 e^{2 x}-c_3 e^x-c_3\right )-c_1 \left (-3 e^{2 x}+e^{3 x}+1\right )\right ) \\ \text {y2}(x)\to e^{-x} \left (-\left (c_1 \left (2 e^x+1\right ) \left (e^x-1\right )^2\right )+2 c_2 e^{3 x}-(c_2+c_3) e^{2 x}+c_3\right ) \\ \text {y3}(x)\to e^{-x} \left (-\left (c_1 \left (-3 e^{2 x}+e^{3 x}+2\right )\right )+c_2 e^{3 x}-(c_2+c_3) e^{2 x}+2 c_3\right ) \\ \end{align*}