74.7.50 problem 53

Internal problem ID [16056]
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 : 53
Date solved : Monday, March 31, 2025 at 02:37:10 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 28

ode:=t*diff(y(t),t)-y(t)-1 = diff(y(t),t)^2-diff(y(t),t); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {1}{4} t^{2}+\frac {1}{2} t -\frac {3}{4} \\ y &= -c_1^{2}+t c_1 +c_1 -1 \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 35

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

\begin{align*} y(t)\to c_1 (t+1)-1-c_1{}^2 \\ y(t)\to \frac {1}{4} \left (t^2+2 t-3\right ) \\ \end{align*}

✓ Sympy. Time used: 2.101 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \frac {t^{2}}{4} + \frac {t}{2} - \frac {\left (C_{1} + t\right )^{2}}{4} - \frac {3}{4} \]