76.4.3 problem 3

Internal problem ID [17397]
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 : 3
Date solved : Tuesday, January 28, 2025 at 10:04:35 AM
CAS classification : [_exact, _rational]

\begin{align*} 3 x^{2}-2 y x +2+\left (6 y^{2}-x^{2}+3\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 368

dsolve((3*x^2-2*x*y(x)+2)+(6*y(x)^2-x^2+3)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{2}/{3}}+6 x^{2}-18}{6 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{1}/{3}}} \\ y &= \frac {\left (-i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{1}/{3}}}{12}+\frac {\left (i \sqrt {3}-1\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{1}/{3}}} \\ y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{2}/{3}}}{6}+\left (-i \sqrt {3}-1\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.751 (sec). Leaf size: 421

DSolve[(3*x^2-2*x*y[x]+2)+(6*y[x]^2-x^2+3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {x^2-3}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}-\frac {\sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}}{6^{2/3}} \\ y(x)\to \frac {\sqrt [3]{6} \left (1+i \sqrt {3}\right ) \left (x^2-3\right )+\left (1-i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\ y(x)\to \frac {\sqrt [3]{6} \left (1-i \sqrt {3}\right ) \left (x^2-3\right )+\left (1+i \sqrt {3}\right ) \left (9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {-2 \left (x^2-3\right )^3+27 \left (x^3+2 x+c_1\right ){}^2}+18 x+9 c_1}} \\ \end{align*}