Internal problem ID [11231]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first.
Article 27. Clairaut equation. Page 56
Problem number: Ex 6.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\left (y^{2}+x^{2}\right ) \left (1+y^{\prime }\right )^{2}-2 \left (x +y\right ) \left (1+y^{\prime }\right ) \left (x +y y^{\prime }\right )+\left (x +y y^{\prime }\right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.547 (sec). Leaf size: 106
dsolve((x^2+y(x)^2)*(1+diff(y(x),x))^2-2*(x+y(x))*(1+diff(y(x),x))*(x+y(x)*diff(y(x),x))+(x+y(x)*diff(y(x),x))^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 y \left (x \right ) = \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 \left (x \right ) = \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: 7.379 (sec). Leaf size: 167
DSolve[(x^2+y[x]^2)*(1+y'[x])^2-2*(x+y[x])*(1+y'[x])*(x+y[x]*y'[x])+(x+y[x]*y'[x])^2==0,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*}