77.1.155 problem 182 (page 297)

Internal problem ID [17974]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 182 (page 297)
Date solved : Monday, March 31, 2025 at 04:52:56 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.206 (sec). Leaf size: 91

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

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

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 1161

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 4.240 (sec). Leaf size: 76

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

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