40.6.10 problem 19

Internal problem ID [6690]
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
Date solved : Monday, January 27, 2025 at 02:23:57 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 103

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 )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +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: 4.222 (sec). Leaf size: 167

DSolve[(D[y[x],x]^2+1)*(x-y[x])^2==(x+y[x]*D[y[x],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*}