74.6.51 problem 58

Internal problem ID [16074]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 58
Date solved : Tuesday, January 28, 2025 at 08:36:03 AM
CAS classification : [_exact]

\begin{align*} -1+{\mathrm e}^{t y} y+y \cos \left (t y\right )+\left (1+{\mathrm e}^{t y} t +t \cos \left (t y\right )\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.108 (sec). Leaf size: 27

dsolve((-1+exp(t*y(t))*y(t)+y(t)*cos(t*y(t)))+(1+exp(t*y(t))*t+t*cos(t*y(t)))*diff(y(t),t)=0,y(t), singsol=all)
 

\[ y = \frac {\operatorname {RootOf}\left (t \sin \left (\textit {\_Z} \right )+t \,{\mathrm e}^{\textit {\_Z}}+c_{1} t -t^{2}+\textit {\_Z} \right )}{t} \]

✓ Solution by Mathematica

Time used: 0.553 (sec). Leaf size: 107

DSolve[(-1+Exp[t*y[t]]*y[t]+y[t]*Cos[t*y[t]])+(1+Exp[t*y[t]]*t+t*Cos[t*y[t]])*D[y[t],t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^t\left (e^{K[1] y(t)} y(t)+\cos (K[1] y(t)) y(t)-1\right )dK[1]+\int _1^{y(t)}\left (e^{t K[2]} t+\cos (t K[2]) t-\int _1^t\left (\cos (K[1] K[2])+e^{K[1] K[2]}+e^{K[1] K[2]} K[1] K[2]-K[1] K[2] \sin (K[1] K[2])\right )dK[1]+1\right )dK[2]=c_1,y(t)\right ] \]