31.5.4 problem 4

Internal problem ID [5747]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:12:30 PM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} u^{\prime }+b u^{2}&=\frac {c}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 37

dsolve(diff(u(x),x)+b*u(x)^2=c*x^(-4),u(x), singsol=all)
 

\[ u \left (x \right ) = \frac {-\sqrt {-b c}\, \tan \left (\frac {\sqrt {-b c}\, \left (c_{1} x -1\right )}{x}\right )+x}{b \,x^{2}} \]

✓ Solution by Mathematica

Time used: 0.343 (sec). Leaf size: 109

DSolve[D[u[x],x]+b*u[x]^2==x^(-4),u[x],x,IncludeSingularSolutions -> True]
 

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