29.31.23 problem 922

Internal problem ID [5502]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 922
Date solved : Monday, January 27, 2025 at 11:32:57 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 84

dsolve((a^2+x^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+b+y(x)^2 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.394 (sec). Leaf size: 100

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

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