60.1.317 problem 318

Internal problem ID [10331]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 318
Date solved : Monday, January 27, 2025 at 07:13:12 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (3 x y^{3}-4 y x +y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 28

dsolve((3*x*y(x)^3-4*x*y(x)+y(x))*diff(y(x),x)+y(x)^2*(y(x)^2-2) = 0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ x +\frac {1}{y^{2}}-\frac {c_{1}}{\sqrt {y^{2}-2}\, y^{2}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.613 (sec). Leaf size: 161

DSolve[y[x]^2*(-2 + y[x]^2) + (y[x] - 4*x*y[x] + 3*x*y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ \text {Solve}\left [x&=\exp \left (\int _1^{y(x)}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right ) \int _1^{y(x)}\frac {\exp \left (-\int _1^{K[2]}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right )}{2 K[2]-K[2]^3}dK[2]+c_1 \exp \left (\int _1^{y(x)}\frac {3 K[1]^2-4}{2 K[1]-K[1]^3}dK[1]\right ),y(x)\right ] \\ y(x)\to 0 \\ y(x)\to -\sqrt {2} \\ y(x)\to \sqrt {2} \\ \end{align*}