76.1.12 problem 12

Internal problem ID [17311]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 09:59:32 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 16

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

\[ y = \frac {1}{1+{\mathrm e}^{-\frac {x^{2}}{2}} c_{1}} \]

✓ Solution by Mathematica

Time used: 0.264 (sec). Leaf size: 46

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [-\frac {x^2}{2}+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}