Internal problem ID [3912]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 15.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {y-y^{\prime } x -a x \sqrt {\left (y^{\prime }\right )^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 223
dsolve(y(x)=x*diff(y(x),x)+a*x*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} x -\frac {{\mathrm e}^{\frac {\arcsinh \left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \relax (x )^{2}}\, a +y \relax (x )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \relax (x )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \relax (x )^{2}}\, a y \relax (x )+x^{2}+y \relax (x )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} = 0 \\ x -\frac {{\mathrm e}^{-\frac {\arcsinh \left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \relax (x )^{2}}\, a -y \relax (x )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {-\frac {a^{2} x^{2}-a^{2} y \relax (x )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \relax (x )^{2}}\, a y \relax (x )-x^{2}-y \relax (x )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.857 (sec). Leaf size: 283
DSolve[y[x]==x*y'[x]+a*x*Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {a \left (\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-a-\frac {i y(x)}{x}+1\right )+\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a-\frac {i y(x)}{x}-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-\frac {i y(x)}{x}\right )\right )}{a^2-1}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {a \left (\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-a-\frac {i y(x)}{x}-1\right )+\log \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a-\frac {i y(x)}{x}+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {a^2-\frac {y(x)^2}{x^2}-1}-\frac {i y(x)}{x}\right )\right )}{a^2-1}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}