71.18.18 problem 16

Internal problem ID [14592]
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 : 16
Date solved : Tuesday, January 28, 2025 at 06:44:00 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 70

dsolve([diff(y__1(x),x)=0*y__1(x)+1*y__2(x)+0*y__3(x)+1*y__4(x),diff(y__2(x),x)=1*y__1(x)+0*y__2(x)-1*y__3(x)+0*y__4(x),diff(y__3(x),x)=0*y__1(x)+0*y__2(x)+0*y__3(x)+1*y__4(x),diff(y__4(x),x)=0*y__1(x)+0*y__2(x)+1*y__3(x)+0*y__4(x)],singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} \\ y_{2} \left (x \right ) &= -c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} +c_{3} {\mathrm e}^{-x}-c_4 \,{\mathrm e}^{x} \\ y_{3} \left (x \right ) &= c_{3} {\mathrm e}^{-x}+c_4 \,{\mathrm e}^{x} \\ y_{4} \left (x \right ) &= -c_{3} {\mathrm e}^{-x}+c_4 \,{\mathrm e}^{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 148

DSolve[{D[ y1[x],x]==0*y1[x]+1*y2[x]+0*y3[x]+1*y4[x],D[ y2[x],x]==1*y1[x]+0*y2[x]-1*y3[x]+0*y4[x],D[ y3[x],x]==0*y1[x]+0*y2[x]+0*y3[x]+1*y4[x],D[ y4[x],x]==0*y1[x]+0*y2[x]+1*y3[x]+0*y4[x]},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}+1\right )+(c_2+c_4) \left (e^{2 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}-1\right )+c_2 e^{2 x}-c_3 e^{2 x}+c_2+c_3\right ) \\ \text {y3}(x)\to \frac {1}{2} e^{-x} \left (c_3 \left (e^{2 x}+1\right )+c_4 \left (e^{2 x}-1\right )\right ) \\ \text {y4}(x)\to \frac {1}{2} e^{-x} \left (c_3 \left (e^{2 x}-1\right )+c_4 \left (e^{2 x}+1\right )\right ) \\ \end{align*}