74.7.52 problem 55

Internal problem ID [16129]
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 : Tuesday, January 28, 2025 at 08:50:22 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Solution by Maple

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

dsolve(1-2*(t*diff(y(t),t)-y(t))=1/diff(y(t),t)^2,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*}

✓ Solution by Mathematica

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

DSolve[1-2*(t*D[y[t],t]-y[t])==1/D[y[t],t]^2,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*}