71.18.14 problem 12

Internal problem ID [14588]
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 : 12
Date solved : Tuesday, January 28, 2025 at 06:43:53 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 41

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

\begin{align*} y_{1} \left (x \right ) &= c_{2} +c_{3} {\mathrm e}^{6 x} \\ y_{2} \left (x \right ) &= c_{2} +c_{3} {\mathrm e}^{6 x}+c_{1} \\ y_{3} \left (x \right ) &= 2 c_{3} {\mathrm e}^{6 x}-c_{2} -\frac {c_{1}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 114

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

\begin{align*} \text {y1}(x)\to \frac {1}{6} \left (c_1 \left (e^{6 x}+5\right )+(c_2+2 c_3) \left (e^{6 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{6} \left (c_1 \left (e^{6 x}-1\right )+c_2 \left (e^{6 x}+5\right )+2 c_3 \left (e^{6 x}-1\right )\right ) \\ \text {y3}(x)\to \frac {1}{3} \left (c_1 \left (e^{6 x}-1\right )+c_2 \left (e^{6 x}-1\right )+c_3 \left (2 e^{6 x}+1\right )\right ) \\ \end{align*}