53.1.16 problem 16

Internal problem ID [8450]
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 : 16
Date solved : Monday, January 27, 2025 at 04:01:40 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 85

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

\begin{align*} y &= -x -\sqrt {x^{2}+2 c_{1}} \\ y &= -x +\sqrt {x^{2}+2 c_{1}} \\ y &= \frac {-c_{1} x -\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y &= \frac {-c_{1} x +\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.818 (sec). Leaf size: 172

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

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