31.5.3 problem 3

Internal problem ID [5746]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 3
Date solved : Monday, January 27, 2025 at 01:12:30 PM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} u^{\prime }+u^{2}&=\frac {c}{x^{{4}/{3}}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 39

dsolve(diff(u(x),x)+u(x)^2=c*x^(-4/3),u(x), singsol=all)
 

\[ u \left (x \right ) = -\frac {3 c}{x^{{1}/{3}} \left (3 \tan \left (3 \sqrt {-c}\, \left (x^{{1}/{3}}-c_{1} \right )\right ) x^{{1}/{3}} \sqrt {-c}+1\right )} \]

✓ Solution by Mathematica

Time used: 0.275 (sec). Leaf size: 183

DSolve[D[u[x],x]+u[x]^2==c*x^(-4/3),u[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} u(x)\to \frac {3 c \left (3 i \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )+8 c_1 \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )}{\sqrt [3]{x} \left (\left (9 i \sqrt {c} \sqrt [3]{x}-8 c_1\right ) \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )+3 \left (8 \sqrt {c} c_1 \sqrt [3]{x}-i\right ) \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )} \\ u(x)\to -\frac {3 c \cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )}{\sqrt [3]{x} \left (\cosh \left (3 \sqrt {c} \sqrt [3]{x}\right )-3 \sqrt {c} \sqrt [3]{x} \sinh \left (3 \sqrt {c} \sqrt [3]{x}\right )\right )} \\ \end{align*}