74.7.37 problem 37

Internal problem ID [16035]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 37
Date solved : Thursday, March 13, 2025 at 07:22:05 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

With initial conditions

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

✓ Maple. Time used: 1.082 (sec). Leaf size: 21

ode:=t*diff(y(t),t)-y(t)-(t^2+y(t)^2)^(1/2) = 0; 
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: 0.299 (sec). Leaf size: 14

ode=t*D[y[t],t]-(y[t]+Sqrt[t^2+y[t]^2])==0; 
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: 1.137 (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)
 

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