23.4.4 problem 8(d)

Internal problem ID [4169]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(d)
Date solved : Sunday, March 30, 2025 at 02:41:30 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 32

ode:=[diff(y__1(x),x) = 4*y__2(x), diff(y__2(x),x) = 4*y__2(x)-y__1(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (c_2 x +c_1 \right ) \\ y_{2} \left (x \right ) &= \frac {{\mathrm e}^{2 x} \left (2 c_2 x +2 c_1 +c_2 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[y1[x],x]==4*y2[x],D[y2[x],x]==4*y2[x]-y1[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to e^{2 x} (-2 c_1 x+4 c_2 x+c_1) \\ \text {y2}(x)\to e^{2 x} (c_1 (-x)+2 c_2 x+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 42

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

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