66.1.41 problem Problem 55

Internal problem ID [13888]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 55
Date solved : Tuesday, January 28, 2025 at 06:07:27 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Solution by Maple

Time used: 0.411 (sec). Leaf size: 101

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

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.496 (sec). Leaf size: 185

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

\begin{align*} y(x)\to -\frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\ \end{align*}