66.1.5 problem Problem 5

Internal problem ID [13852]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 5
Date solved : Tuesday, January 28, 2025 at 06:05:19 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.163 (sec). Leaf size: 49

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

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

✓ Solution by Mathematica

Time used: 0.514 (sec). Leaf size: 68

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

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