68.4.19 problem 19

Internal problem ID [17199]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 19
Date solved : Thursday, October 02, 2025 at 01:51:05 PM
CAS classification : [_separable]

\begin{align*} 3 \sin \left (t \right )-\sin \left (3 t \right )&=\left (\cos \left (4 y\right )-4 \cos \left (y\right )\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 243

ode:=3*sin(t)-sin(3*t) = (cos(4*y(t))-4*cos(y(t)))*diff(y(t),t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \arctan \left (\frac {4 \cos \left (t \right )^{3}-12 c_1 -12 \cos \left (t \right )}{6 \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_1 \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_1^{2}+288 c_1 \cos \left (t \right )+135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )^{3}-3 \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_1 \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_1^{2}+288 c_1 \cos \left (t \right )+135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )-12}, \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_1 \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_1^{2}+288 c_1 \cos \left (t \right )+135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )\right ) \]

✓ Mathematica. Time used: 0.36 (sec). Leaf size: 45

ode=3*Sin[t]-Sin[3*t]==(Cos[4*y[t]]-4*Cos[y[t]])*D[y[t],t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}(4 \cos (K[1])-\cos (4 K[1]))dK[1]\&\right ]\left [\int _1^t-4 \sin ^3(K[2])dK[2]+c_1\right ] \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((4*cos(y(t)) - cos(4*y(t)))*Derivative(y(t), t) + 3*sin(t) - sin(3*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out