74.7.49 problem 52

Internal problem ID [16126]
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 : 52
Date solved : Tuesday, January 28, 2025 at 08:50:18 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 53

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

\begin{align*} y &= \frac {-7 t^{2}-2 t +1}{8 t} \\ y &= c_{1} t -\frac {1}{4}-\frac {\sqrt {-8 c_{1} -7}}{4} \\ y &= c_{1} t -\frac {1}{4}+\frac {\sqrt {-8 c_{1} -7}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.275 (sec). Leaf size: 90

DSolve[t*D[y[t],t]-y[t]-2*(t*D[y[t],t]-y[t])^2==D[y[t],t]+1,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {1}{8} \left (-7 t-e^{-2 c_1} t-2+2 e^{-c_1}\right ) \\ y(t)\to \frac {1}{8} \left (-7 t-e^{-2 c_1} t-2-2 e^{-c_1}\right ) \\ y(t)\to \frac {1}{8} (-7 t-2) \\ y(t)\to \frac {1}{8} \left (-7 t+\frac {1}{t}-2\right ) \\ \end{align*}