13.3.7 problem 7

Internal problem ID [2324]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.4. Page 24
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 01:54:24 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {2 t}{y+t^{2} y} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=3 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 20

ode:=diff(y(t),t) = 2*t/(y(t)+t^2*y(t)); 
ic:=y(2) = 3; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \sqrt {9-2 \ln \left (5\right )+2 \ln \left (t^{2}+1\right )} \]

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 23

ode=D[y[t],t] == 2*t/(y[t]+t^2*y[t]); 
ic=y[2]==3; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \sqrt {2 \log \left (t^2+1\right )+9-2 \log (5)} \]

✓ Sympy. Time used: 0.510 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t/(t**2*y(t) + y(t)) + Derivative(y(t), t),0) 
ics = {y(2): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \sqrt {2 \log {\left (t^{2} + 1 \right )} - \log {\left (25 \right )} + 9} \]