17.4.12 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 : Tuesday, September 30, 2025 at 07:06:23 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.201 (sec). Leaf size: 79

ode:=[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)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 127

ode={D[y1[x],x]==y2[x],D[y2[x],x]==-y1[x]+y3[x],D[y3[x],x]==-y2[x]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__2(x) + Derivative(y__1(x), x),0),Eq(y__1(x) - y__3(x) + Derivative(y__2(x), x),0),Eq(y__2(x) + Derivative(y__3(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
 

\[ \left [ y^{1}{\left (x \right )} = C_{1} + C_{2} \sin {\left (\sqrt {2} x \right )} - C_{3} \cos {\left (\sqrt {2} x \right )}, \ y^{2}{\left (x \right )} = \sqrt {2} C_{2} \cos {\left (\sqrt {2} x \right )} + \sqrt {2} C_{3} \sin {\left (\sqrt {2} x \right )}, \ y^{3}{\left (x \right )} = C_{1} - C_{2} \sin {\left (\sqrt {2} x \right )} + C_{3} \cos {\left (\sqrt {2} x \right )}\right ] \]