76.5.20 problem 20

Internal problem ID [17440]
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 2.7 (Substitution Methods). Problems at page 108
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 10:37:39 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

dsolve(diff(y(t),t)=r*y(t)-k^2*y(t)^2,y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 49

DSolve[D[y[t],t]==r*y[t]-k^2*y[t]^2,y[t],t,IncludeSingularSolutions -> True]
 

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