50.4.21 problem 21

Internal problem ID [7870]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page 20
Problem number : 21
Date solved : Monday, January 27, 2025 at 03:29:08 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \frac {4 y^{2}-2 x^{2}}{4 x y^{2}-x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-x^{2} y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.640 (sec). Leaf size: 157

dsolve(( (4*y(x)^2-2*x^2)/(4*x*y(x)^2-x^3))+( (8*y(x)^2-x^2)/(4*y(x)^3-x^2*y(x)) )*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {\sqrt {2}\, \sqrt {c_{1} x \left (c_{1}^{3} x^{3}-\sqrt {c_{1}^{6} x^{6}+16}\right )}}{4 x \,c_{1}^{2}} \\ y &= \frac {\sqrt {2}\, \sqrt {c_{1} x \left (c_{1}^{3} x^{3}-\sqrt {c_{1}^{6} x^{6}+16}\right )}}{4 x \,c_{1}^{2}} \\ y &= -\frac {\sqrt {2}\, \sqrt {c_{1} x \left (c_{1}^{3} x^{3}+\sqrt {c_{1}^{6} x^{6}+16}\right )}}{4 x \,c_{1}^{2}} \\ y &= \frac {\sqrt {2}\, \sqrt {c_{1} x \left (c_{1}^{3} x^{3}+\sqrt {c_{1}^{6} x^{6}+16}\right )}}{4 x \,c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 12.795 (sec). Leaf size: 297

DSolve[( (4*y[x]^2-2*x^2)/(4*x*y[x]^2-x^3))+( (8*y[x]^2-x^2)/(4*y[x]^3-x^2*y[x]) )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-16 e^{2 c_1}}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{2 \sqrt {2}} \\ \end{align*}