76.5.16 problem 16

Internal problem ID [17436]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:35:41 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} t^{2} y^{\prime }+2 t y-y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 60

dsolve(t^2*diff(y(t),t)+2*t*y(t)-y(t)^3=0,y(t), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {5}\, \sqrt {5 c_{1} t^{6}+2 t}}{5 c_{1} t^{5}+2} \\ y &= -\frac {\sqrt {5}\, \sqrt {5 c_{1} t^{6}+2 t}}{5 c_{1} t^{5}+2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.561 (sec). Leaf size: 63

DSolve[t^2*D[y[t],t]+2*t*y[t]-y[t]^3==0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\frac {\sqrt {5} \sqrt {t}}{\sqrt {2+5 c_1 t^5}} \\ y(t)\to \frac {\sqrt {5} \sqrt {t}}{\sqrt {2+5 c_1 t^5}} \\ y(t)\to 0 \\ \end{align*}