31.6.11 problem 11

Internal problem ID [5760]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 11
Date solved : Monday, January 27, 2025 at 01:12:58 PM
CAS classification : [_quadrature]

\begin{align*} 1+{y^{\prime }}^{2}&=\frac {\left (x +a \right )^{2}}{2 a x +x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 42

dsolve(1+(diff(y(x),x))^2=(x+a)^2/(x^2+2*a*x),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= a \ln \left (x +a +\sqrt {x \left (x +2 a \right )}\right )+c_{1} \\ y \left (x \right ) &= -a \ln \left (x +a +\sqrt {x \left (x +2 a \right )}\right )+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.218 (sec). Leaf size: 103

DSolve[1+(D[y[x],x])^2==(x+a)^2/(x^2+2*a*x),y[x],x,IncludeSingularSolutions -> True]
 

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