76.3.14 problem 14

Internal problem ID [17314]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 14
Date solved : Monday, March 31, 2025 at 03:52:01 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y^{\prime }&=-\frac {t}{2}+\frac {\sqrt {t^{2}+4 y}}{2} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=-1 \end{align*}

✓ Maple. Time used: 0.572 (sec). Leaf size: 17

ode:=diff(y(t),t) = -1/2*t+1/2*(t^2+4*y(t))^(1/2); 
ic:=y(2) = -1; 
dsolve([ode,ic],y(t), singsol=all);
 

\begin{align*} y &= 1-t \\ y &= -\frac {t^{2}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.791 (sec). Leaf size: 10

ode=D[y[t],t]==(-t+(t^2+4*y[t])^(1/2))/2; 
ic={y[2]==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 1-t \]

✓ Sympy. Time used: 0.761 (sec). Leaf size: 5

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t/2 - sqrt(t**2 + 4*y(t))/2 + Derivative(y(t), t),0) 
ics = {y(2): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 1 - t \]