76.4.19 problem 25

Internal problem ID [17413]
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 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 25
Date solved : Tuesday, January 28, 2025 at 10:06:23 AM
CAS classification : [[_homogeneous, `class D`], _rational]

\begin{align*} 3 x^{2} y+2 y x +y^{3}+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.193 (sec). Leaf size: 293

dsolve((3*x^2*y(x)+2*x*y(x)+y(x)^3)+ (x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {2^{{1}/{3}} \left (x^{2} c_{1}^{2} {\mathrm e}^{6 x}-\frac {2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{2}/{3}}}{2}\right ) {\mathrm e}^{-3 x}}{{\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{1}/{3}} c_{1}} \\ y &= -\frac {2^{{1}/{3}} \left (2 \left (i \sqrt {3}-1\right ) x^{2} {\mathrm e}^{6 x} c_{1}^{2}+2^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{2}/{3}}\right ) {\mathrm e}^{-3 x}}{4 {\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{1}/{3}} c_{1}} \\ y &= \frac {2^{{1}/{3}} \left (2 x^{2} \left (1+i \sqrt {3}\right ) {\mathrm e}^{6 x} c_{1}^{2}+2^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{2}/{3}}\right ) {\mathrm e}^{-3 x}}{4 {\left (\left (1+\sqrt {4 x^{6} c_{1}^{2} {\mathrm e}^{6 x}+1}\right ) c_{1}^{2} {\mathrm e}^{6 x}\right )}^{{1}/{3}} c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 46

DSolve[(3*x^2*y[x]+2*x*y[x]+y[x]^3) + (x^2+y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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