65.17.3 problem 4

Internal problem ID [15846]
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 : 4
Date solved : Thursday, October 02, 2025 at 10:28:37 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )+2 y_{2} \left (x \right )+x -1\\ y_{2}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+2 y_{2} \left (x \right )-5 x -2 \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=-2 \\ y_{2} \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.138 (sec). Leaf size: 17

ode:=[diff(y__1(x),x) = y__1(x)+2*y__2(x)+x-1, diff(y__2(x),x) = 3*y__1(x)+2*y__2(x)-5*x-2]; 
ic:=[y__1(0) = -2, y__2(0) = 3]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 479

ode={D[ y1[x],x]==y1[x]+2*y2[x]+x-1,D[ y2[x],x]==3*y1[x]+2*y2[x]-5*x-2}; 
ic={y1[0]==-2,y2[0]==3}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)&\to \frac {1}{5} e^{-x} \left (-\left (\left (2 e^{5 x}+3\right ) \int _1^0\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]\right )+\left (2 e^{5 x}+3\right ) \int _1^x\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]+2 \left (-\left (\left (e^{5 x}-1\right ) \int _1^0-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]\right )+\left (e^{5 x}-1\right ) \int _1^x-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+e^{5 x}-6\right )\right )\\ \text {y2}(x)&\to \frac {1}{5} e^{-x} \left (-3 \left (e^{5 x}-1\right ) \int _1^0\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]+3 \left (e^{5 x}-1\right ) \int _1^x\frac {1}{5} e^{-4 K[1]} \left (-8 K[1]+e^{5 K[1]} (13 K[1]+1)-6\right )dK[1]-3 e^{5 x} \int _1^0-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+3 e^{5 x} \int _1^x-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+2 \int _1^x-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]-2 \int _1^0-\frac {1}{5} e^{-4 K[2]} \left (12 K[2]+e^{5 K[2]} (13 K[2]+1)+9\right )dK[2]+3 e^{5 x}+12\right ) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-x - y__1(x) - 2*y__2(x) + Derivative(y__1(x), x) + 1,0),Eq(5*x - 3*y__1(x) - 2*y__2(x) + Derivative(y__2(x), x) + 2,0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

\[ \left [ y^{1}{\left (x \right )} = - C_{1} e^{- x} + \frac {2 C_{2} e^{4 x}}{3} + 3 x - 2, \ y^{2}{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{4 x} - 2 x + 3\right ] \]