61.2.18 problem 18

Internal problem ID [12024]
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
Date solved : Monday, January 27, 2025 at 11:51:19 PM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} x^{4} y^{\prime }&=-x^{4} y^{2}-a^{2} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 24

dsolve(x^4*diff(y(x),x)=-x^4*y(x)^2-a^2,y(x), singsol=all)
 

\[ y = \frac {-a \tan \left (\frac {a \left (c_{1} x -1\right )}{x}\right )+x}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.519 (sec). Leaf size: 94

DSolve[x^4*D[y[x],x]==-x^4*y[x]^2-a^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 a^2 c_1 e^{\frac {2 i a}{x}}+i a \left (e^2+2 c_1 x e^{\frac {2 i a}{x}}\right )+e^2 x}{x^2 \left (e^2+2 i a c_1 e^{\frac {2 i a}{x}}\right )} \\ y(x)\to \frac {x-i a}{x^2} \\ \end{align*}