60.1.429 problem 431

Internal problem ID [10443]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 431
Date solved : Monday, January 27, 2025 at 07:43:58 PM
CAS classification : [_separable]

\begin{align*} x^{2} {y^{\prime }}^{2}-y^{4}+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 52

dsolve(x^2*diff(y(x),x)^2-y(x)^4+y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y &= -1 \\ y &= 1 \\ y &= 0 \\ y &= \csc \left (-\ln \left (x \right )+c_{1} \right ) \operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_{1} \right )\right ) \\ y &= -\csc \left (-\ln \left (x \right )+c_{1} \right ) \operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_{1} \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.535 (sec). Leaf size: 88

DSolve[y[x]^2 - y[x]^4 + x^2*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {\sec ^2(-\log (x)+c_1)} \\ y(x)\to \sqrt {\sec ^2(-\log (x)+c_1)} \\ y(x)\to -\sqrt {\sec ^2(\log (x)+c_1)} \\ y(x)\to \sqrt {\sec ^2(\log (x)+c_1)} \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}