88.11.8 problem 9

Internal problem ID [24055]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 54
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:55:16 PM
CAS classification : system_of_ODEs

\begin{align*} y^{\prime }&=y+z \left (x \right )+x\\ z^{\prime }\left (x \right )&=1-y-z \left (x \right ) \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 40

ode:=[diff(y(x),x) = x+y(x)+z(x), diff(z(x),x) = 1-y(x)-z(x)]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= \frac {1}{6} x^{3}+x^{2}+c_1 x +c_2 \\ z \left (x \right ) &= -\frac {1}{2} x^{2}+x +c_1 -\frac {1}{6} x^{3}-c_1 x -c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 54

ode={D[y[x],x]==x+y[x]+z[x],D[z[x],x]==1-y[x]-z[x]}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^3}{6}+x^2+(c_1+c_2) x+c_1\\ z(x)&\to -\frac {x^3}{6}-\frac {x^2}{2}-(-1+c_1+c_2) x+c_2 \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x - y(x) - z(x) + Derivative(y(x), x),0),Eq(y(x) + z(x) + Derivative(z(x), x) - 1,0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + C_{2} x + C_{2} + \frac {x^{3}}{6} + x^{2}, \ z{\left (x \right )} = - C_{1} - \frac {x^{3}}{6} - \frac {x^{2}}{2} - x \left (C_{2} - 1\right )\right ] \]