15.5.10 problem 10

Internal problem ID [2946]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 9, page 38
Problem number : 10
Date solved : Monday, January 27, 2025 at 07:00:05 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 18

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

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

✓ Solution by Mathematica

Time used: 3.146 (sec). Leaf size: 285

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

\[ \text {Solve}\left [\frac {\left (2-\frac {2 \left (x^3+2 x y(x)\right )}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}\right ) \left (\frac {x^3+2 x y(x)}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}+2\right ) \left (\left (1-\frac {x \left (x^2+2 y(x)\right )}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}\right ) \log \left (\frac {2-\frac {2 \left (x^3+2 x y(x)\right )}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {x^3+2 x y(x)}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}-1\right ) \log \left (\frac {\frac {2 \left (x^3+2 x y(x)\right )}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}+4}{\sqrt [3]{2}}\right )-3\right )}{9 \sqrt [3]{2} \left (-\frac {\left (x^2+2 y(x)\right )^3}{\left (x^2-y(x)\right )^3}+\frac {3 \left (x^3+2 x y(x)\right )}{\sqrt [3]{x^3} \left (x^2-y(x)\right )}-2\right )}=\frac {2\ 2^{2/3} x \log (x)}{9 \sqrt [3]{x^3}}+c_1,y(x)\right ] \]