74.7.52 problem 55

Internal problem ID [16050]
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 : 55
Date solved : Thursday, March 13, 2025 at 07:36:42 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.075 (sec). Leaf size: 60

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

\begin{align*} y &= \frac {3 t^{{2}/{3}}}{2}-\frac {1}{2} \\ y &= -\frac {3 t^{{2}/{3}}}{4}-\frac {3 i \sqrt {3}\, t^{{2}/{3}}}{4}-\frac {1}{2} \\ y &= -\frac {3 t^{{2}/{3}}}{4}+\frac {3 i \sqrt {3}\, t^{{2}/{3}}}{4}-\frac {1}{2} \\ y &= -\frac {1}{2}+c_{1} t +\frac {1}{2 c_{1}^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 93

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

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

✗ Sympy

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

Timed Out