60.1.324 problem 325

Internal problem ID [10338]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 325
Date solved : Monday, January 27, 2025 at 07:13:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 3.404 (sec). Leaf size: 122

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

\[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 y\right ) \sqrt {3}}{3 x}\right )}{7}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}+4 x^{2} y+2 x y^{2}+2 y^{3}\right )}{3 x^{3}}\right )}{7}-\frac {4 \ln \left (2\right )}{7}-\frac {2 \ln \left (\frac {x^{4}+x^{3} y+3 x^{2} y^{2}+x y^{3}+y^{4}}{x^{4}}\right )}{7}+\frac {\ln \left (\frac {y-x}{x}\right )}{7}-\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.169 (sec). Leaf size: 58

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1] \left (K[1]^3-2\right )}{(K[1]-1) \left (K[1]^4+K[1]^3+3 K[1]^2+K[1]+1\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]