71.18.10 problem 8

Internal problem ID [14584]
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 : 8
Date solved : Tuesday, January 28, 2025 at 06:43:49 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=5 y_{1} \left (x \right )-5 y_{2} \left (x \right )-5 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+4 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )-5 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.474 (sec). Leaf size: 70

dsolve([diff(y__1(x),x)=5*y__1(x)-5*y__2(x)-5*y__3(x),diff(y__2(x),x)=-1*y__1(x)+4*y__2(x)+2*y__3(x),diff(y__3(x),x)=3*y__1(x)-5*y__2(x)-3*y__3(x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 109

DSolve[{D[ y1[x],x]==5*y1[x]-5*y2[x]-5*y3[x],D[ y2[x],x]==-1*y1[x]+4*y2[x]+2*y3[x],D[ y3[x],x]==3*y1[x]-5*y2[x]-3*y3[x]},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to e^{2 x} (c_1 \cos (x)+(3 c_1-5 (c_2+c_3)) \sin (x)) \\ \text {y2}(x)\to e^{2 x} (-c_1 (\sin (x)+\cos (x)-1)+c_3 (2 \sin (x)+\cos (x)-1)+c_2 (2 \sin (x)+\cos (x))) \\ \text {y3}(x)\to e^{2 x} (c_1 \cos (x)+(3 c_1-5 (c_2+c_3)) \sin (x)-c_1+c_3) \\ \end{align*}