77.1.155 problem 182 (page 297)

Internal problem ID [18045]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 182 (page 297)
Date solved : Tuesday, January 28, 2025 at 08:28:30 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.302 (sec). Leaf size: 87

dsolve([diff(y(x),x)=z(x)^2/y(x),diff(z(x),x)=y(x)^2/z(x)],singsol=all)
 

\begin{align*} \left \{y &= -\frac {{\mathrm e}^{-2 x} \sqrt {-2 \,{\mathrm e}^{2 x} \left ({\mathrm e}^{4 x} c_{1} -c_{2} \right )}}{2}, y = \frac {{\mathrm e}^{-2 x} \sqrt {-2 \,{\mathrm e}^{2 x} \left ({\mathrm e}^{4 x} c_{1} -c_{2} \right )}}{2}\right \} \\ \left \{z \left (x \right ) &= \sqrt {y y^{\prime }}, z \left (x \right ) = -\sqrt {y y^{\prime }}\right \} \\ \end{align*}

✓ Solution by Mathematica

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

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

\begin{align*} z(x)\to -i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to -\frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to -i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to -\frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to -i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to \frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to -i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to \frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to -\frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to -\frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to \frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to i \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x-c_2))} \\ y(x)\to \frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {-\tanh (2 (x-c_2))}}{\sqrt [4]{\text {sech}^2(2 (x-c_2))}} \\ z(x)\to -\sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to -\frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to -\sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to -\frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to -\sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to \frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to -\sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to \frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to -\frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to -\frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to \frac {i \sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ z(x)\to \sqrt {2} \sqrt [4]{c_1 \cosh ^2(2 (x+c_2))} \\ y(x)\to \frac {\sqrt {2} \sqrt [4]{c_1} \sqrt {\tanh (2 (x+c_2))}}{\sqrt [4]{\text {sech}^2(2 (x+c_2))}} \\ \end{align*}