Internal problem ID [9605]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Riccati, _special]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime }+x^{4} y^{2}+a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve(x^4*diff(y(x),x)=-x^4*y(x)^2-a^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\sqrt {a^{2}}\, \tan \left (\frac {\sqrt {a^{2}}\, \left (c_{1} x -1\right )}{x}\right )-x}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.26 (sec). Leaf size: 116
DSolve[x^4*y'[x]==-x^4*y[x]^2-a^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (x+i \sqrt {-a^2} c_1\right ) \cos \left (\frac {a}{x}\right )+\frac {\left (a^2-i \sqrt {-a^2} c_1 x\right ) \sin \left (\frac {a}{x}\right )}{a}}{x^2 \left (\cos \left (\frac {a}{x}\right )+\frac {i \sqrt {a} c_1 \sin \left (\frac {a}{x}\right )}{\sqrt {-a}}\right )} \\ y(x)\to \frac {x-a \cot \left (\frac {a}{x}\right )}{x^2} \\ \end{align*}