83.31.6 problem 6

Internal problem ID [19398]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (E) at page 112
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 01:37:20 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 1+{y^{\prime }}^{2}+y y^{\prime \prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 39

dsolve(1+diff(y(x),x)^2+y(x)*diff(y(x),x$2)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 11.473 (sec). Leaf size: 83

DSolve[1+D[y[x],x]^2+y[x]*D[y[x],{x,2}]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2} \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2} \\ y(x)\to -\sqrt {-(x+c_2){}^2} \\ y(x)\to \sqrt {-(x+c_2){}^2} \\ \end{align*}