Internal problem ID [5039]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems.
page 12
Problem number: Example 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _Bernoulli]
Solve \begin {gather*} \boxed {2 x y^{\prime }+\left (y^{4} x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 67
dsolve(2*x*diff(y(x),x)+(x^2*y(x)^4+1)*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {1}{\sqrt {\sqrt {2 \ln \relax (x )+c_{1}}\, x}} \\ y \relax (x ) = \frac {1}{\sqrt {-\sqrt {2 \ln \relax (x )+c_{1}}\, x}} \\ y \relax (x ) = -\frac {1}{\sqrt {\sqrt {2 \ln \relax (x )+c_{1}}\, x}} \\ y \relax (x ) = -\frac {1}{\sqrt {-\sqrt {2 \ln \relax (x )+c_{1}}\, x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.325 (sec). Leaf size: 92
DSolve[2*x*y'[x]+(x^2*y[x]^4+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to -\frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {i}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to \frac {1}{\sqrt [4]{x^2 (2 \log (x)+c_1)}} \\ y(x)\to 0 \\ \end{align*}