Internal problem ID [8893]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 558.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {a x \sqrt {{y^{\prime }}^{2}+1}+x y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 340
dsolve(a*x*(diff(y(x),x)^2+1)^(1/2)+x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} \frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{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 {-a^{2} x^{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 {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{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 {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{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 {-a^{2} x^{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 {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{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: 1.026 (sec). Leaf size: 223
DSolve[-y[x] + x*y'[x] + a*x*Sqrt[1 + y'[x]^2]==0,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*}