31.6.8 problem 8

Internal problem ID [5757]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 8
Date solved : Monday, January 27, 2025 at 01:12:52 PM
CAS classification : [_quadrature]

\begin{align*} x&=\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 113

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

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

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 154

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

\begin{align*} y(x)\to \frac {\left (a^2-1\right )^{3/2} \left (-\sqrt {\frac {a^2+x^2-1}{a^2-1}}\right ) \text {arcsinh}\left (\frac {x}{\sqrt {a^2-1}}\right )-x \left (-a x \sqrt {a^2+x^2-1}+a^2+x^2-1\right )}{2 \left (a^2-1\right ) \sqrt {a^2+x^2-1}}+c_1 \\ y(x)\to \frac {1}{2} \left (\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}+\log \left (\sqrt {a^2+x^2-1}+x\right )\right )+c_1 \\ \end{align*}