14.2.14 problem 15

Internal problem ID [2502]
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 : 15
Date solved : Tuesday, March 04, 2025 at 02:13:29 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.207 (sec). Leaf size: 21

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

\begin{align*} y &= -\frac {t^{2}}{2}+\frac {1}{2} \\ y &= \frac {t^{2}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.331 (sec). Leaf size: 14

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

\[ y(t)\to \frac {1}{2} \left (t^2-1\right ) \]

✓ Sympy. Time used: 1.134 (sec). Leaf size: 8

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

\[ y{\left (t \right )} = t \sinh {\left (\log {\left (t \right )} \right )} \]