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 : Monday, January 27, 2025 at 05:54:45 AM
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*}

✓ Solution by Maple

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

dsolve([t*diff(y(t),t)=y(t)+sqrt(t^2+y(t)^2),y(1) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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