77.1.149 problem 176 (page 265)

Internal problem ID [17968]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 176 (page 265)
Date solved : Monday, March 31, 2025 at 04:52:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )&=y \left (x \right )+z \left (x \right )\\ \frac {d}{d x}z \left (x \right )&=y \left (x \right )+z \left (x \right )+x \end{align*}

✓ Maple. Time used: 0.190 (sec). Leaf size: 44

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

\begin{align*} y \left (x \right ) &= -\frac {x^{2}}{4}+\frac {{\mathrm e}^{2 x} c_1}{2}-\frac {x}{4}+c_2 \\ z \left (x \right ) &= -\frac {x}{4}+\frac {{\mathrm e}^{2 x} c_1}{2}-\frac {1}{4}+\frac {x^{2}}{4}-c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 84

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

\begin{align*} y(x)\to \frac {1}{8} \left (-2 x^2-2 x+4 c_1 \left (e^{2 x}+1\right )+4 c_2 e^{2 x}-1-4 c_2\right ) \\ z(x)\to \frac {1}{8} \left (2 x^2-2 x+4 c_1 \left (e^{2 x}-1\right )+4 c_2 e^{2 x}-1+4 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = - C_{1} + C_{2} e^{2 x} - \frac {x^{2}}{4} - \frac {x}{4} - \frac {1}{8}, \ z{\left (x \right )} = C_{1} + C_{2} e^{2 x} + \frac {x^{2}}{4} - \frac {x}{4} - \frac {1}{8}\right ] \]