Internal problem ID [10358]
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]]
\[ \boxed {x^{4} y^{\prime }+x^{4} y^{2}=-a^{2}} \]
✓ 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)
\[ y \left (x \right ) = -\frac {\sqrt {a^{2}}\, \tan \left (\frac {\sqrt {a^{2}}\, \left (x c_{1} -1\right )}{x}\right )-x}{x^{2}} \]
✓ Solution by Mathematica
Time used: 1.107 (sec). Leaf size: 87
DSolve[x^4*y'[x]==-x^4*y[x]^2-a^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 i a^2 c_1 e^{\frac {2 i a}{x}}+2 a c_1 x e^{\frac {2 i a}{x}}+a-i x}{x^2 \left (2 a c_1 e^{\frac {2 i a}{x}}-i\right )} y(x)\to \frac {x-i a}{x^2} \end{align*}