Internal problem ID [4420]
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]
\[ \boxed {y-x y^{\prime }-a x \sqrt {1+{y^{\prime }}^{2}}=0} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 340
dsolve(y(x)=x*diff(y(x),x)+a*x*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} \frac {x \sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.993 (sec). Leaf size: 223
DSolve[y[x]==x*y'[x]+a*x*Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}