Internal problem ID [3817]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 37
Problem number: 1129.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\sqrt {\left (a \,x^{2}+y^{2}\right ) \left (1+\left (y^{\prime }\right )^{2}\right )}-y y^{\prime }-a x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.711 (sec). Leaf size: 180
dsolve(((a*x^2+y(x)^2)*(1+diff(y(x),x)^2))^(1/2)-y(x)*diff(y(x),x)-a*x = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \sqrt {-a}\, x \\ y \relax (x ) = -\sqrt {-a}\, x \\ y \relax (x ) = \frac {-x^{\frac {\sqrt {\left (a -1\right ) a}+a}{a}} a^{3}+x^{-\frac {\sqrt {\left (a -1\right ) a}-a}{a}} c_{1}^{2}+x^{\frac {\sqrt {\left (a -1\right ) a}+a}{a}} a^{2}}{2 \sqrt {\left (a -1\right ) a}\, c_{1}} \\ y \relax (x ) = -\frac {x^{-\frac {\sqrt {\left (a -1\right ) a}-a}{a}} a^{3}-x^{-\frac {\sqrt {\left (a -1\right ) a}-a}{a}} a^{2}-x^{\frac {\sqrt {\left (a -1\right ) a}+a}{a}} c_{1}^{2}}{2 \sqrt {\left (a -1\right ) a}\, c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.739 (sec). Leaf size: 241
DSolve[((a x^2+y[x]^2)(1+(y'[x])^2))^(1/2) -y[x] y'[x]-a x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (x^{2 \sqrt {\frac {a-1}{a}}}-e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) \\ y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) \\ y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right ) \\ \end{align*}