14.2.15 problem 16

Internal problem ID [2503]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 16
Date solved : Sunday, March 30, 2025 at 12:04:01 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=2*t*y(t)*diff(y(t),t) = 3*y(t)^2-t^2; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \sqrt {c_1 t +1}\, t \\ y &= -\sqrt {c_1 t +1}\, t \\ \end{align*}

✓ Mathematica. Time used: 0.201 (sec). Leaf size: 127

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

\begin{align*} y(t)\to \frac {i t \left (c_1 \operatorname {BesselJ}\left (1,\frac {1}{2} i \sqrt {3} t\right )-\operatorname {BesselY}\left (1,-\frac {1}{2} i \sqrt {3} t\right )\right )}{\sqrt {3} \left (\operatorname {BesselY}\left (0,-\frac {1}{2} i \sqrt {3} t\right )+c_1 \operatorname {BesselJ}\left (0,\frac {1}{2} i \sqrt {3} t\right )\right )} \\ y(t)\to \frac {i t \operatorname {BesselJ}\left (1,\frac {1}{2} i \sqrt {3} t\right )}{\sqrt {3} \operatorname {BesselJ}\left (0,\frac {1}{2} i \sqrt {3} t\right )} \\ \end{align*}

✓ Sympy. Time used: 0.375 (sec). Leaf size: 26

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

\[ \left [ y{\left (t \right )} = - t \sqrt {C_{1} t + 1}, \ y{\left (t \right )} = t \sqrt {C_{1} t + 1}\right ] \]