Internal problem ID [5449]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\frac {4 y^{2}-2 x^{2}}{4 y^{2} x -x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-x^{2} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.844 (sec). Leaf size: 225
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 \relax (x ) = -\frac {c_{1} x -\frac {2 x^{2} c_{1}^{2}-\sqrt {2 x^{4} c_{1}^{4}-2 c_{1} \sqrt {x^{6} c_{1}^{6}+16}\, x}}{2 x c_{1}}}{2 c_{1}} \\ y \relax (x ) = -\frac {c_{1} x -\frac {2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}-2 c_{1} \sqrt {x^{6} c_{1}^{6}+16}\, x}}{2 x c_{1}}}{2 c_{1}} \\ y \relax (x ) = -\frac {c_{1} x -\frac {2 x^{2} c_{1}^{2}-\sqrt {2 x^{4} c_{1}^{4}+2 c_{1} \sqrt {x^{6} c_{1}^{6}+16}\, x}}{2 x c_{1}}}{2 c_{1}} \\ y \relax (x ) = -\frac {c_{1} x -\frac {2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}+2 c_{1} \sqrt {x^{6} c_{1}^{6}+16}\, x}}{2 x c_{1}}}{2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.152 (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]) )*y'[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*}