Internal problem ID [5347]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page
74
Problem number: 19.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\left ({y^{\prime }}^{2}+1\right ) \left (x -y\right )^{2}-\left (x +y y^{\prime }\right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 106
dsolve((diff(y(x),x)^2+1)*(x-y(x))^2=(x+y(x)*diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} y = 0 y = \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 \textit {\_a}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x y = \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}-2 \textit {\_a}^{2}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \end{align*}
✓ Solution by Mathematica
Time used: 4.36 (sec). Leaf size: 167
DSolve[(y'[x]^2+1)*(x-y[x])^2==(x+y[x]*y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} y(x)\to \sqrt {-x \left (x+2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} y(x)\to e^{\frac {c_1}{2}}-\sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )} y(x)\to \sqrt {x \left (-x+2 e^{\frac {c_1}{2}}\right )}+e^{\frac {c_1}{2}} y(x)\to -\sqrt {-x^2} y(x)\to \sqrt {-x^2} \end{align*}