82.54.18 problem Ex. 18

Internal problem ID [19034]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 18
Date solved : Tuesday, January 28, 2025 at 12:46:15 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 39

dsolve(y(x)*diff(y(x),x$2)+diff(y(x),x)^2+1=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: 13.021 (sec). Leaf size: 83

DSolve[y[x]*D[y[x],{x,2}]+D[y[x],x]^2+1==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*}