68.8.34 problem 34

Internal problem ID [17457]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 34
Date solved : Thursday, October 02, 2025 at 02:23:46 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=\pi \\ \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 18

ode:=y(t)^2+(2*t*y(t)-2*cos(y(t))*sin(y(t)))*diff(y(t),t) = 0; 
ic:=[y(0) = Pi]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (t \,\textit {\_Z}^{2}+2 \cos \left (\textit {\_Z} \right )-2\right )}{2} \]

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 42

ode=(y[t]^2)+(2*t*y[t]-2*Cos[y[t]]*Sin[y[t]])*D[y[t],t]==0; 
ic={y[0]==Pi}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [t=\frac {\int _0^{y(t)}\sin (2 K[1])dK[1]}{y(t)^2}-\frac {\int _0^{\pi }\sin (2 K[1])dK[1]}{y(t)^2},y(t)\right ] \]

✓ Sympy. Time used: 1.541 (sec). Leaf size: 19

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

\[ t y^{2}{\left (t \right )} + \frac {\cos {\left (2 y{\left (t \right )} \right )}}{2} - \frac {1}{2} = 0 \]