34.6.1 problem 1

Internal problem ID [6075]
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
Date solved : Monday, January 27, 2025 at 01:34:56 PM
CAS classification : [_rational, _Riccati]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 36

dsolve(diff(y(x),x)+y(x)^2=a^2/x^4,y(x), singsol=all)
 

\[ y = \frac {-\sqrt {-a^{2}}\, \tan \left (\frac {\sqrt {-a^{2}}\, \left (c_1 x -1\right )}{x}\right )+x}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.399 (sec). Leaf size: 81

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

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