77.1.25 problem 41 (page 55)

Internal problem ID [17915]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 41 (page 55)
Date solved : Tuesday, January 28, 2025 at 11:12:22 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Riccati, _special]]

\begin{align*} y^{\prime }&=\frac {y^{2}}{3}+\frac {2}{3 x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 165

dsolve(diff(y(x),x)=1/3*y(x)^2+2/(3*x^2),y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (c_{1}^{2} x \right )^{{1}/{3}}+\frac {x c_{1}}{\left (c_{1}^{2} x \right )^{{1}/{3}}}-c_{1} +2 x}{\left (-x +c_{1} \right ) x} \\ y &= -\frac {\left (-i \sqrt {3}-1\right ) \left (c_{1}^{2} x \right )^{{2}/{3}}+\left (-2 c_{1} +4 x \right ) \left (c_{1}^{2} x \right )^{{1}/{3}}+x c_{1} \left (i \sqrt {3}-1\right )}{2 \left (c_{1}^{2} x \right )^{{1}/{3}} \left (x -c_{1} \right ) x} \\ y &= \frac {\left (1-i \sqrt {3}\right ) \left (c_{1}^{2} x \right )^{{2}/{3}}+\left (2 c_{1} -4 x \right ) \left (c_{1}^{2} x \right )^{{1}/{3}}+x c_{1} \left (1+i \sqrt {3}\right )}{2 \left (c_{1}^{2} x \right )^{{1}/{3}} \left (x -c_{1} \right ) x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.136 (sec). Leaf size: 38

DSolve[D[y[x],x]==1/3*y[x]^2+2/(3*x^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {2 \sqrt [3]{x}+c_1}{x^{4/3}+c_1 x} \\ y(x)\to -\frac {1}{x} \\ \end{align*}