76.5.33 problem 34

Internal problem ID [17453]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 34
Date solved : Tuesday, January 28, 2025 at 10:38:45 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

Solution by Maple

Time used: 0.075 (sec). Leaf size: 283

dsolve(diff(x(y),y)=(2*x(y)*y+x(y)^2)/(3*y^2+2*x(y)*y),x(y), singsol=all)
 
\begin{align*} x \left (y \right ) &= \frac {12^{{1}/{3}} \left (y 12^{{1}/{3}} c_{1} +{\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{2}/{3}}\right )}{6 c_{1} {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}}} \\ x \left (y \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} y \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1} \right )}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\ x \left (y \right ) &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} y \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) c_{1} \right ) 3^{{1}/{3}}}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\ \end{align*}

Solution by Mathematica

Time used: 0.134 (sec). Leaf size: 40

DSolve[D[x[y],y]==(2*x[y]*y+x[y]^2)/(3*y^2+2*x[y]*y),x[y],y,IncludeSingularSolutions -> True]
 
\[ \text {Solve}\left [\int _1^{\frac {x(y)}{y}}\frac {2 K[1]+3}{K[1] (K[1]+1)}dK[1]=-\log (y)+c_1,x(y)\right ] \]