29.37.11 problem 1129

Internal problem ID [5669]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1129
Date solved : Monday, January 27, 2025 at 01:03:13 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \sqrt {\left (a \,x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y^{\prime } y-a x&=0 \end{align*}

✓ Solution by Maple

Time used: 1.530 (sec). Leaf size: 142

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 \left (x \right ) &= x \sqrt {-a} \\ y \left (x \right ) &= -x \sqrt {-a} \\ y \left (x \right ) &= -\frac {a^{2} \left (a -1\right ) x^{\frac {-\sqrt {\left (a -1\right ) a}+a}{a}}-x^{\frac {a +\sqrt {\left (a -1\right ) a}}{a}} c_{1}^{2}}{2 \sqrt {\left (a -1\right ) a}\, c_{1}} \\ y \left (x \right ) &= -\frac {-x^{\frac {-\sqrt {\left (a -1\right ) a}+a}{a}} c_{1}^{2}+x^{\frac {a +\sqrt {\left (a -1\right ) a}}{a}} a^{2} \left (a -1\right )}{2 \sqrt {\left (a -1\right ) a}\, c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.444 (sec). Leaf size: 113

DSolve[((a*x^2+y[x]^2)*(1+(D[y[x],x])^2))^(1/2) -y[x]*D[y[x],x]-a*x==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-a x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {a-1}{a}}+1}-\frac {1}{2} a e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \\ \end{align*}