77.1.150 problem 177 (page 265)

Internal problem ID [18040]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 177 (page 265)
Date solved : Tuesday, January 28, 2025 at 08:28:29 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 24

dsolve([diff(y(x),x)=y(x)^2/z(x),diff(z(x),x)=1/2*y(x)],singsol=all)
 

\begin{align*} \left \{z \left (x \right ) &= -\frac {1}{c_{1} x +c_{2}}\right \} \\ \{y &= 2 z^{\prime }\left (x \right )\} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 45

DSolve[{D[y[x],x]==y[x]^2/z[x],D[z[x],x]==1/2*y[x]},{y[x],z[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} z(x)\to 2 c_1 \sqrt {\frac {c_1{}^2}{(x+c_1 c_2){}^2}} \\ y(x)\to \frac {4 c_1{}^2}{(x+c_1 c_2){}^2} \\ \end{align*}