6.1 problem 1

Internal problem ID [4223]

Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section: Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number: 1.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_rational, _Riccati]

Solve \begin {gather*} \boxed {y^{\prime }+y^{2}-\frac {a^{2}}{x^{4}}=0} \end {gather*}

Solution by Maple

Time used: 0.004 (sec). Leaf size: 38

dsolve(diff(y(x),x)+y(x)^2=a^2/x^4,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: 0.228 (sec). Leaf size: 128

DSolve[y'[x]+y[x]^2==a^2/x^4,y[x],x,IncludeSingularSolutions -> True]
 

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