71.18.11 problem 9

Internal problem ID [14585]
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 : 9
Date solved : Tuesday, January 28, 2025 at 06:43:50 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=4 y_{1} \left (x \right )+6 y_{2} \left (x \right )+6 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+3 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=-y_{1} \left (x \right )-4 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end{align*}

Solution by Maple

Time used: 0.124 (sec). Leaf size: 62

dsolve([diff(y__1(x),x)=4*y__1(x)+6*y__2(x)+6*y__3(x),diff(y__2(x),x)=1*y__1(x)+3*y__2(x)+2*y__3(x),diff(y__3(x),x)=-1*y__1(x)-4*y__2(x)-3*y__3(x)],singsol=all)
 
\begin{align*} y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{-x}+c_{3} {\mathrm e}^{4 x} \\ y_{2} \left (x \right ) &= \frac {c_{2} {\mathrm e}^{-x}}{3}+\frac {c_{3} {\mathrm e}^{4 x}}{3}+{\mathrm e}^{x} c_{1} \\ y_{3} \left (x \right ) &= -\frac {7 c_{2} {\mathrm e}^{-x}}{6}-\frac {c_{3} {\mathrm e}^{4 x}}{3}-{\mathrm e}^{x} c_{1} \\ \end{align*}

Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 145

DSolve[{D[ y1[x],x]==4*y1[x]+6*y2[x]+6*y3[x],D[ y2[x],x]==1*y1[x]+3*y2[x]+2*y3[x],D[ y3[x],x]==-1*y1[x]-4*y2[x]-3*y3[x]},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions -> True]
 
\begin{align*} \text {y1}(x)\to \frac {1}{5} e^{-x} \left ((5 c_1+6 (c_2+c_3)) e^{5 x}-6 (c_2+c_3)\right ) \\ \text {y2}(x)\to \frac {1}{15} e^{-x} \left (-5 (c_1-3 c_2) e^{2 x}+(5 c_1+6 (c_2+c_3)) e^{5 x}-6 (c_2+c_3)\right ) \\ \text {y3}(x)\to \frac {1}{3} (c_1-3 c_2) e^x+\frac {7}{5} (c_2+c_3) e^{-x}-\frac {1}{15} (5 c_1+6 (c_2+c_3)) e^{4 x} \\ \end{align*}