77.1.156 problem 183 (page 297)

Internal problem ID [17975]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 183 (page 297)
Date solved : Monday, March 31, 2025 at 04:52:57 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )&=\frac {y \left (x \right )^{2}}{z \left (x \right )}\\ \frac {d}{d x}z \left (x \right )&=\frac {z \left (x \right )^{2}}{y \left (x \right )} \end{align*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 36

ode:=[diff(y(x),x) = y(x)^2/z(x), diff(z(x),x) = z(x)^2/y(x)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 269

ode={D[y[x],x]==y[x]^2/z[x],D[z[x],x]==z[x]^2/y[x]}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} z(x)\to -\frac {\sqrt {\frac {1}{1+\cosh (2 x-2 c_2)}}}{\sqrt {c_1} \sqrt {\tanh ^2(x-c_2)}} \\ y(x)\to \frac {\sqrt {\frac {1}{1+\cosh (2 x-2 c_2)}}}{\sqrt {c_1}} \\ z(x)\to \frac {\sqrt {\frac {1}{1+\cosh (2 x-2 c_2)}}}{\sqrt {c_1} \sqrt {\tanh ^2(x-c_2)}} \\ y(x)\to -\frac {\sqrt {\frac {1}{1+\cosh (2 x-2 c_2)}}}{\sqrt {c_1}} \\ z(x)\to -\frac {\sqrt {\text {sech}^2(x+c_2)}}{\sqrt {2} \sqrt {c_1} \sqrt {\tanh ^2(x+c_2)}} \\ y(x)\to -\frac {\sqrt {\text {sech}^2(x+c_2)}}{\sqrt {2} \sqrt {c_1}} \\ z(x)\to \frac {\sqrt {\text {sech}^2(x+c_2)}}{\sqrt {2} \sqrt {c_1} \sqrt {\tanh ^2(x+c_2)}} \\ y(x)\to \frac {\sqrt {\text {sech}^2(x+c_2)}}{\sqrt {2} \sqrt {c_1}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-y(x)**2/z(x) + Derivative(y(x), x),0),Eq(Derivative(z(x), x) - z(x)**2/y(x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)