65.18.10 problem 8

Internal problem ID [15871]
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 : 8
Date solved : Thursday, October 02, 2025 at 10:28:56 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=5 y_{1} \left (x \right )-5 y_{2} \left (x \right )-5 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+4 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )-5 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.260 (sec). Leaf size: 70

ode:=[diff(y__1(x),x) = 5*y__1(x)-5*y__2(x)-5*y__3(x), diff(y__2(x),x) = -y__1(x)+4*y__2(x)+2*y__3(x), diff(y__3(x),x) = 3*y__1(x)-5*y__2(x)-3*y__3(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_2 +\cos \left (x \right ) c_3 \right ) \\ y_{2} \left (x \right ) &= -\frac {\left (2 \sin \left (x \right ) c_2 -\sin \left (x \right ) c_3 +\cos \left (x \right ) c_2 +2 \cos \left (x \right ) c_3 -5 c_1 \right ) {\mathrm e}^{2 x}}{5} \\ y_{3} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_2 +\cos \left (x \right ) c_3 -c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 109

ode={D[ y1[x],x]==5*y1[x]-5*y2[x]-5*y3[x],D[ y2[x],x]==-1*y1[x]+4*y2[x]+2*y3[x],D[ y3[x],x]==3*y1[x]-5*y2[x]-3*y3[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)&\to e^{2 x} (c_1 \cos (x)+(3 c_1-5 (c_2+c_3)) \sin (x))\\ \text {y2}(x)&\to e^{2 x} (-c_1 (\sin (x)+\cos (x)-1)+c_3 (2 \sin (x)+\cos (x)-1)+c_2 (2 \sin (x)+\cos (x)))\\ \text {y3}(x)&\to e^{2 x} (c_1 \cos (x)+(3 c_1-5 (c_2+c_3)) \sin (x)-c_1+c_3) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-5*y__1(x) + 5*y__2(x) + 5*y__3(x) + Derivative(y__1(x), x),0),Eq(y__1(x) - 4*y__2(x) - 2*y__3(x) + Derivative(y__2(x), x),0),Eq(-3*y__1(x) + 5*y__2(x) + 3*y__3(x) + Derivative(y__3(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
 

\[ \left [ y^{1}{\left (x \right )} = - C_{1} e^{2 x} \sin {\left (x \right )} + C_{2} e^{2 x} \cos {\left (x \right )}, \ y^{2}{\left (x \right )} = - C_{3} e^{2 x} + \left (\frac {C_{1}}{5} - \frac {2 C_{2}}{5}\right ) e^{2 x} \cos {\left (x \right )} + \left (\frac {2 C_{1}}{5} + \frac {C_{2}}{5}\right ) e^{2 x} \sin {\left (x \right )}, \ y^{3}{\left (x \right )} = - C_{1} e^{2 x} \sin {\left (x \right )} + C_{2} e^{2 x} \cos {\left (x \right )} + C_{3} e^{2 x}\right ] \]