34.2.9 problem 9

Internal problem ID [6039]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 9
Date solved : Monday, January 27, 2025 at 01:34:13 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 44

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

\[ y = x^{-\frac {2}{a}} {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (-\frac {x^{-\frac {2}{a}} {\mathrm e}^{-\frac {2 c_1}{a}}}{a}\right )-2 c_1}{a}} \]

✓ Solution by Mathematica

Time used: 60.019 (sec). Leaf size: 29

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

\[ y(x)\to -a W\left (-\frac {e^{\frac {c_1}{a}} x^{-2/a}}{a}\right ) \]