28.4.6 problem 7.6

Internal problem ID [4538]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.6
Date solved : Monday, January 27, 2025 at 09:23:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )-y_{2} \left (x \right )&=0\\ 4 y_{1} \left (x \right )+\frac {d}{d x}y_{2} \left (x \right )-4 y_{2} \left (x \right )-2 y_{3} \left (x \right )&=0\\ -2 y_{1} \left (x \right )+y_{2} \left (x \right )+\frac {d}{d x}y_{3} \left (x \right )+y_{3} \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 43

dsolve([diff(y__1(x),x)-y__2(x)=0,4*y__1(x)+diff(y__2(x),x)-4*y__2(x)-2*y__3(x)=0, -2*y__1(x)+y__2(x)+diff(y__3(x),x)+y__3(x)=0],singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= c_{1} +c_{2} {\mathrm e}^{x}+c_3 \,{\mathrm e}^{2 x} \\ y_{2} \left (x \right ) &= c_{2} {\mathrm e}^{x}+2 c_3 \,{\mathrm e}^{2 x} \\ y_{3} \left (x \right ) &= 2 c_{1} +\frac {c_{2} {\mathrm e}^{x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 119

DSolve[{D[y1[x],x]-y2[x]==0,4*y1[x]+D[y2[x],x]-4*y2[x]-2*y3[x]==0, -2*y1[x]+y2[x]+D[y3[x],x]+y3[x]==0},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to c_1 \left (4 e^x-2 e^{2 x}-1\right )+\frac {1}{2} \left (e^x-1\right ) \left (c_2 \left (3 e^x-1\right )+2 c_3 \left (e^x-1\right )\right ) \\ \text {y2}(x)\to e^x \left (-4 c_1 \left (e^x-1\right )+c_2 \left (3 e^x-2\right )+2 c_3 \left (e^x-1\right )\right ) \\ \text {y3}(x)\to 2 c_1 \left (e^x-1\right )-c_2 \left (e^x-1\right )-c_3 \left (e^x-2\right ) \\ \end{align*}