74.8.19 problem 19

Internal problem ID [16084]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 19
Date solved : Monday, March 31, 2025 at 02:41:18 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} t \ln \left (y\right )+\left (\frac {t^{2}}{2 y}+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.094 (sec). Leaf size: 31

ode:=t*ln(y(t))+(1/2*t^2/y(t)+1)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.965 (sec). Leaf size: 31

ode=(t*Log[y[t]])+(t^2/(2*y[t])+1)*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \frac {1}{2} t^2 W\left (\frac {2 e^{\frac {c_1}{t^2}}}{t^2}\right ) \\ y(t)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.787 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*log(y(t)) + (t**2/(2*y(t)) + 1)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2} W\left (\frac {2 e^{\frac {C_{1}}{t^{2}}}}{t^{2}}\right )}{2} \]