53.1.14 problem 14

Internal problem ID [8448]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 14
Date solved : Monday, January 27, 2025 at 04:01:27 PM
CAS classification : [_quadrature]

\begin{align*} x y {y^{\prime }}^{2}+\left (x y^{2}-1\right ) y^{\prime }-y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.119 (sec). Leaf size: 57

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

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