71.18.1 problem 1

Internal problem ID [14496]
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 : 1
Date solved : Thursday, March 13, 2025 at 03:31:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+5 \,{\mathrm e}^{x}\\ \frac {d}{d x}y_{2} \left (x \right )&=y_{1} \left (x \right )+4 y_{2} \left (x \right )-2 \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.811 (sec). Leaf size: 111

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

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2} +{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1} +\frac {{\mathrm e}^{-x}}{3}-\frac {5 \,{\mathrm e}^{x}}{2} \\ y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2}}{3}+\frac {{\mathrm e}^{3 x} \sqrt {2}\, \sin \left (\sqrt {2}\, x \right ) c_{2}}{3}-\frac {{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1}}{3}-\frac {{\mathrm e}^{3 x} \sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) c_{1}}{3}+\frac {{\mathrm e}^{-x}}{3}+\frac {5 \,{\mathrm e}^{x}}{6} \\ \end{align*}

✓ Mathematica. Time used: 1.323 (sec). Leaf size: 420

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

\begin{align*} \text {y1}(x)\to -\frac {1}{2} e^{3 x} \left (\left (\sqrt {2} \sin \left (\sqrt {2} x\right )-2 \cos \left (\sqrt {2} x\right )\right ) \int _1^x\left (5 e^{-2 K[1]} \cos \left (\sqrt {2} K[1]\right )+\frac {e^{-4 K[1]} \left (-6+5 e^{2 K[1]}\right ) \sin \left (\sqrt {2} K[1]\right )}{\sqrt {2}}\right )dK[1]+3 \sqrt {2} \sin \left (\sqrt {2} x\right ) \int _1^x\frac {1}{2} e^{-4 K[2]} \left (-4 \cos \left (\sqrt {2} K[2]\right )-\sqrt {2} \left (-2+5 e^{2 K[2]}\right ) \sin \left (\sqrt {2} K[2]\right )\right )dK[2]-2 c_1 \cos \left (\sqrt {2} x\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} x\right )+3 \sqrt {2} c_2 \sin \left (\sqrt {2} x\right )\right ) \\ \text {y2}(x)\to \frac {1}{2} e^{3 x} \left (\sqrt {2} \sin \left (\sqrt {2} x\right ) \int _1^x\left (5 e^{-2 K[1]} \cos \left (\sqrt {2} K[1]\right )+\frac {e^{-4 K[1]} \left (-6+5 e^{2 K[1]}\right ) \sin \left (\sqrt {2} K[1]\right )}{\sqrt {2}}\right )dK[1]+\left (\sqrt {2} \sin \left (\sqrt {2} x\right )+2 \cos \left (\sqrt {2} x\right )\right ) \int _1^x\frac {1}{2} e^{-4 K[2]} \left (-4 \cos \left (\sqrt {2} K[2]\right )-\sqrt {2} \left (-2+5 e^{2 K[2]}\right ) \sin \left (\sqrt {2} K[2]\right )\right )dK[2]+2 c_2 \cos \left (\sqrt {2} x\right )+\sqrt {2} c_1 \sin \left (\sqrt {2} x\right )+\sqrt {2} c_2 \sin \left (\sqrt {2} x\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.691 (sec). Leaf size: 207

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

\[ \left [ y^{1}{\left (x \right )} = \left (C_{1} - \sqrt {2} C_{2}\right ) e^{3 x} \sin {\left (\sqrt {2} x \right )} - \left (\sqrt {2} C_{1} + C_{2}\right ) e^{3 x} \cos {\left (\sqrt {2} x \right )} - \frac {5 e^{x} \sin ^{2}{\left (\sqrt {2} x \right )}}{2} - \frac {5 e^{x} \cos ^{2}{\left (\sqrt {2} x \right )}}{2} + \frac {e^{- x} \sin ^{2}{\left (\sqrt {2} x \right )}}{3} + \frac {e^{- x} \cos ^{2}{\left (\sqrt {2} x \right )}}{3}, \ y^{2}{\left (x \right )} = - C_{1} e^{3 x} \sin {\left (\sqrt {2} x \right )} + C_{2} e^{3 x} \cos {\left (\sqrt {2} x \right )} + \frac {5 e^{x} \sin ^{2}{\left (\sqrt {2} x \right )}}{6} + \frac {5 e^{x} \cos ^{2}{\left (\sqrt {2} x \right )}}{6} + \frac {e^{- x} \sin ^{2}{\left (\sqrt {2} x \right )}}{3} + \frac {e^{- x} \cos ^{2}{\left (\sqrt {2} x \right )}}{3}\right ] \]