71.17.5 problem 6

Internal problem ID [14482]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 7. Systems of First-Order Differential Equations. Exercises page 329
Problem number : 6
Date solved : Friday, March 14, 2025 at 04:47:53 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}-2 x\\ \frac {d}{d x}y_{2} \left (x \right )&=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x}+5 x \end{align*}

With initial conditions

\begin{align*} y_{1} \left (-1\right ) = 3\\ y_{2} \left (-1\right ) = -3 \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 38

ode:=[diff(y__1(x),x) = 5*y__1(x)/x+4*y__2(x)/x-2*x, diff(y__2(x),x) = -6*y__1(x)/x-5*y__2(x)/x+5*x]; 
ic:=y__1(-1) = 3y__2(-1) = -3; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (x \right ) &= \frac {2 x^{3}+x^{2}-2}{x} \\ y_{2} \left (x \right ) &= -\frac {2 x^{3}+2 x^{2}-6}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 33

ode={D[ y1[x],x]==5*y1[x]/x+4*y2[x]/x-2*x,D[ y2[x],x]==-6*y1[x]/x-5*y2[x]/x+5*x}; 
ic={y1[-1]==3,y2[-1]==-3}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to 2 x^2+x-\frac {2}{x} \\ \text {y2}(x)\to -\frac {x^3+x^2-3}{x} \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(2*x + Derivative(y__1(x), x) - 5*y__1(x)/x - 4*y__2(x)/x,0),Eq(-5*x + Derivative(y__2(x), x) + 6*y__1(x)/x + 5*y__2(x)/x,0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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