71.18.16 problem 14

Internal problem ID [14511]
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 : 14
Date solved : Thursday, March 13, 2025 at 03:31:54 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 )&=-3 y_{1} \left (x \right )+2 y_{3} \left (x \right )\\ \frac {d}{d x}y_{3} \left (x \right )&=y_{4} \left (x \right )\\ \frac {d}{d x}y_{4} \left (x \right )&=2 y_{1} \left (x \right )-5 y_{3} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.204 (sec). Leaf size: 547

ode:=[diff(y__1(x),x) = y__2(x), diff(y__2(x),x) = -3*y__1(x)+2*y__3(x), diff(y__3(x),x) = y__4(x), diff(y__4(x),x) = 2*y__1(x)-5*y__3(x)]; 
dsolve(ode);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 730

ode={D[ y1[x],x]==0*y1[x]+1*y2[x]+0*y3[x]+0*y4[x],D[ y2[x],x]==-3*y1[x]+0*y2[x]+2*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]==2*y1[x]+0*y2[x]-5*y3[x]+0*y4[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 0.471 (sec). Leaf size: 442

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
y__4 = Function("y__4") 
ode=[Eq(-y__2(x) + Derivative(y__1(x), x),0),Eq(3*y__1(x) - 2*y__3(x) + Derivative(y__2(x), x),0),Eq(-y__4(x) + Derivative(y__3(x), x),0),Eq(-2*y__1(x) + 5*y__3(x) + Derivative(y__4(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x),y__4(x)],ics=ics)
 

\[ \left [ y^{1}{\left (x \right )} = \frac {C_{1} \left (9 - 5 \sqrt {5}\right ) \sqrt {\sqrt {5} + 4} \sin {\left (x \sqrt {\sqrt {5} + 4} \right )}}{22} + \frac {C_{2} \left (9 - 5 \sqrt {5}\right ) \sqrt {\sqrt {5} + 4} \cos {\left (x \sqrt {\sqrt {5} + 4} \right )}}{22} + \frac {C_{3} \sqrt {4 - \sqrt {5}} \left (9 + 5 \sqrt {5}\right ) \sin {\left (x \sqrt {4 - \sqrt {5}} \right )}}{22} + \frac {C_{4} \sqrt {4 - \sqrt {5}} \left (9 + 5 \sqrt {5}\right ) \cos {\left (x \sqrt {4 - \sqrt {5}} \right )}}{22}, \ y^{2}{\left (x \right )} = \frac {C_{1} \left (1 - \sqrt {5}\right ) \cos {\left (x \sqrt {\sqrt {5} + 4} \right )}}{2} - \frac {C_{2} \left (1 - \sqrt {5}\right ) \sin {\left (x \sqrt {\sqrt {5} + 4} \right )}}{2} + \frac {C_{3} \left (1 + \sqrt {5}\right ) \cos {\left (x \sqrt {4 - \sqrt {5}} \right )}}{2} - \frac {C_{4} \left (1 + \sqrt {5}\right ) \sin {\left (x \sqrt {4 - \sqrt {5}} \right )}}{2}, \ y^{3}{\left (x \right )} = \frac {C_{1} \left (4 - \sqrt {5}\right ) \sqrt {\sqrt {5} + 4} \sin {\left (x \sqrt {\sqrt {5} + 4} \right )}}{11} + \frac {C_{2} \left (4 - \sqrt {5}\right ) \sqrt {\sqrt {5} + 4} \cos {\left (x \sqrt {\sqrt {5} + 4} \right )}}{11} + \frac {C_{3} \sqrt {4 - \sqrt {5}} \left (\sqrt {5} + 4\right ) \sin {\left (x \sqrt {4 - \sqrt {5}} \right )}}{11} + \frac {C_{4} \sqrt {4 - \sqrt {5}} \left (\sqrt {5} + 4\right ) \cos {\left (x \sqrt {4 - \sqrt {5}} \right )}}{11}, \ y^{4}{\left (x \right )} = C_{1} \cos {\left (x \sqrt {\sqrt {5} + 4} \right )} - C_{2} \sin {\left (x \sqrt {\sqrt {5} + 4} \right )} + C_{3} \cos {\left (x \sqrt {4 - \sqrt {5}} \right )} - C_{4} \sin {\left (x \sqrt {4 - \sqrt {5}} \right )}\right ] \]