12.5.37 problem 34

Internal problem ID [1661]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 34
Date solved : Monday, January 27, 2025 at 05:21:18 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 272

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

\begin{align*} y &= \frac {\left (\left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}-12\right ) x}{6 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\ y &= -\frac {\left (i \sqrt {3}\, \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}-12\right ) x}{12 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\ y &= \frac {x \left (\left (i \sqrt {3}-1\right ) \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{2}/{3}}+12 i \sqrt {3}+12\right )}{12 \left (108 \ln \left (x \right )+108 c_1 +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_1 +81 c_1^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 45.333 (sec). Leaf size: 398

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

\begin{align*} y(x)\to \frac {-2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\ y(x)\to \frac {2 \sqrt [3]{3} \left (1+i \sqrt {3}\right ) x^2+i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\ y(x)\to \frac {2 \sqrt [3]{3} \left (1-i \sqrt {3}\right ) x^2-\sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 \log (x)+9 c_1 x^3}} \\ \end{align*}