23.4.12 problem 11(c)

Internal problem ID [4177]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 11(c)
Date solved : Monday, January 27, 2025 at 08:41:20 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.100 (sec). Leaf size: 79

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

\begin{align*} y_{1} \left (x \right ) &= -\frac {c_{2} \sqrt {2}\, \cos \left (\sqrt {2}\, x \right )}{2}+\frac {c_3 \sqrt {2}\, \sin \left (\sqrt {2}\, x \right )}{2}+c_{1} \\ y_{2} \left (x \right ) &= c_{2} \sin \left (\sqrt {2}\, x \right )+c_3 \cos \left (\sqrt {2}\, x \right ) \\ y_{3} \left (x \right ) &= \frac {c_{2} \sqrt {2}\, \cos \left (\sqrt {2}\, x \right )}{2}-\frac {c_3 \sqrt {2}\, \sin \left (\sqrt {2}\, x \right )}{2}+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 127

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

\begin{align*} \text {y1}(x)\to \frac {1}{2} \left ((c_1-c_3) \cos \left (\sqrt {2} x\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} x\right )+c_1+c_3\right ) \\ \text {y2}(x)\to c_2 \cos \left (\sqrt {2} x\right )+\frac {(c_3-c_1) \sin \left (\sqrt {2} x\right )}{\sqrt {2}} \\ \text {y3}(x)\to \frac {1}{2} \left ((c_3-c_1) \cos \left (\sqrt {2} x\right )-\sqrt {2} c_2 \sin \left (\sqrt {2} x\right )+c_1+c_3\right ) \\ \end{align*}