60.1.298 problem 299

Internal problem ID [10312]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 299
Date solved : Monday, January 27, 2025 at 07:07:59 PM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 238

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

\begin{align*} y &= \frac {12^{{1}/{3}} \left (x^{3} 12^{{1}/{3}}+{\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}\right )}{6 x {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}}} \\ y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{3} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{3} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right )\right )}{12 {\left (\left (\sqrt {-12 x^{5}+81 c_{1}^{2}}+9 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 35.373 (sec). Leaf size: 328

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

\begin{align*} y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ \end{align*}