Internal problem ID [4324]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 37
Problem number: 1126.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 340
dsolve(a*x*sqrt(1+diff(y(x),x)^2)+x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} \frac {x \sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{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 {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{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}+y \left (x \right )^{2} a^{2}-2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{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 {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{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: 0.992 (sec). Leaf size: 223
DSolve[a x Sqrt[1+(y'[x])^2]+x y'[x] -y[x]==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*}