75.11.12 problem 271

Internal problem ID [16863]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 271
Date solved : Tuesday, January 28, 2025 at 09:35:43 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.137 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.201 (sec). Leaf size: 119

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

\begin{align*} y(x)\to -\sqrt {-x^2-2 c_1 x+1-c_1{}^2} \\ y(x)\to \sqrt {-x^2-2 c_1 x+1-c_1{}^2} \\ y(x)\to -\sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to \sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}