23.4.11 problem 11(b)

Internal problem ID [4176]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 11(b)
Date solved : Tuesday, March 04, 2025 at 05:54:56 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d x}y_{1} \left (x \right )+\frac {d}{d x}y_{2} \left (x \right )-4 y_{1} \left (x \right )-y_{2} \left (x \right )&={\mathrm e}^{x}\\ \frac {d}{d x}y_{1} \left (x \right )+3 y_{1} \left (x \right )+y_{2} \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 45

ode:=[2*diff(y__1(x),x)+diff(y__2(x),x)-4*y__1(x)-y__2(x) = exp(x), diff(y__1(x),x)+3*y__1(x)+y__2(x) = 0]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= -\frac {3 c_{2} \sin \left (x \right )}{10}-\frac {3 \cos \left (x \right ) c_{1}}{10}-\frac {{\mathrm e}^{x}}{2}+\frac {c_{2} \cos \left (x \right )}{10}-\frac {c_{1} \sin \left (x \right )}{10} \\ y_{2} \left (x \right ) &= c_{2} \sin \left (x \right )+\cos \left (x \right ) c_{1} +2 \,{\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 55

ode={2*D[y1[x],x]+D[y2[x],x]-4*y1[x]-y2[x]==Exp[x],D[y1[x],x]+3*y1[x]+y2[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to -\frac {e^x}{2}+c_1 \cos (x)-(3 c_1+c_2) \sin (x) \\ \text {y2}(x)\to 2 e^x+c_2 \cos (x)+(10 c_1+3 c_2) \sin (x) \\ \end{align*}

✓ Sympy. Time used: 0.233 (sec). Leaf size: 80

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

\[ \left [ y^{1}{\left (x \right )} = - \left (\frac {C_{1}}{10} - \frac {3 C_{2}}{10}\right ) \sin {\left (x \right )} - \left (\frac {3 C_{1}}{10} + \frac {C_{2}}{10}\right ) \cos {\left (x \right )} - \frac {e^{x} \sin ^{2}{\left (x \right )}}{2} - \frac {e^{x} \cos ^{2}{\left (x \right )}}{2}, \ y^{2}{\left (x \right )} = C_{1} \cos {\left (x \right )} - C_{2} \sin {\left (x \right )} + 2 e^{x} \sin ^{2}{\left (x \right )} + 2 e^{x} \cos ^{2}{\left (x \right )}\right ] \]