Internal problem ID [8138]
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]
Solve \begin {gather*} \boxed {a x \sqrt {\left (y^{\prime }\right )^{2}+1}+x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.288 (sec). Leaf size: 223
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*} 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: 1.829 (sec). Leaf size: 395
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 {a \left (-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )+\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \text {ArcTan}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 i \text {ArcTan}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}