57.4.11 problem 4(c)

Internal problem ID [14335]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number : 4(c)
Date solved : Thursday, October 02, 2025 at 09:31:44 AM
CAS classification : [_separable]

\begin{align*} \left (2 u+1\right ) u^{\prime }-t -1&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

ode:=(2*u(t)+1)*diff(u(t),t)-t-1 = 0; 
dsolve(ode,u(t), singsol=all);
 

\begin{align*} u &= -\frac {1}{2}-\frac {\sqrt {2 t^{2}+4 c_1 +4 t +1}}{2} \\ u &= -\frac {1}{2}+\frac {\sqrt {2 t^{2}+4 c_1 +4 t +1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 59

ode=(2*u[t]+1)*D[u[t],t]-(1+t)==0; 
ic={}; 
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
 

\begin{align*} u(t)&\to \frac {1}{2} \left (-1-\sqrt {2 t^2+4 t+1+4 c_1}\right )\\ u(t)&\to \frac {1}{2} \left (-1+\sqrt {2 t^2+4 t+1+4 c_1}\right ) \end{align*}

✓ Sympy. Time used: 0.292 (sec). Leaf size: 42

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

\[ \left [ u{\left (t \right )} = - \frac {\sqrt {C_{1} + 2 t^{2} + 4 t}}{2} - \frac {1}{2}, \ u{\left (t \right )} = \frac {\sqrt {C_{1} + 2 t^{2} + 4 t}}{2} - \frac {1}{2}\right ] \]