65.18.6 problem 5 c

Internal problem ID [15867]
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 : 5 c
Date solved : Thursday, October 02, 2025 at 10:28:54 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+4 x -2\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right )+3 x \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 35

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

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{x} c_2 +{\mathrm e}^{-x} c_1 +x \\ y_{2} \left (x \right ) &= \frac {{\mathrm e}^{x} c_2}{3}+{\mathrm e}^{-x} c_1 -1+2 x \\ \end{align*}

✓ Mathematica. Time used: 0.962 (sec). Leaf size: 333

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

\begin{align*} \text {y1}(x)&\to e^{-2 x} \left (\cos \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]-\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+c_1 \cos \left (\sqrt {3} x\right )-\sqrt {3} c_2 \sin \left (\sqrt {3} x\right )\right )\\ \text {y2}(x)&\to \frac {1}{3} e^{-2 x} \left (3 \cos \left (\sqrt {3} x\right ) \int _1^x\frac {1}{3} e^{2 K[2]} \left (9 \cos \left (\sqrt {3} K[2]\right ) K[2]+2 \sqrt {3} (1-2 K[2]) \sin \left (\sqrt {3} K[2]\right )\right )dK[2]+\sqrt {3} \sin \left (\sqrt {3} x\right ) \int _1^xe^{2 K[1]} \left (\cos \left (\sqrt {3} K[1]\right ) (4 K[1]-2)+3 \sqrt {3} K[1] \sin \left (\sqrt {3} K[1]\right )\right )dK[1]+3 c_2 \cos \left (\sqrt {3} x\right )+\sqrt {3} c_1 \sin \left (\sqrt {3} x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 32

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

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