74.6.40 problem 41

Internal problem ID [16063]
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 : 41
Date solved : Tuesday, January 28, 2025 at 08:34:12 AM
CAS classification : [_exact]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.499 (sec). Leaf size: 25

dsolve([(-2*x-y(x)*cos(x*y(x)))+(2*y(x)-x*cos(x*y(x)))*diff(y(x),x)=0,y(0) = 0],y(x), singsol=all)
 

\[ y = \frac {\operatorname {RootOf}\left (-x^{4}-x^{2} \sin \left (\textit {\_Z} \right )+\textit {\_Z}^{2}\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 75

DSolve[{(-2*x-y[x]*Cos[x*y[x]])+(2*y[x]-x*Cos[x*y[x]])*D[y[x],x]==0,{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _0^x(2 K[1]+\cos (K[1] y(x)) y(x))dK[1]+\int _0^{y(x)}\left (x \cos (x K[2])-2 K[2]-\int _0^x(\cos (K[1] K[2])-K[1] K[2] \sin (K[1] K[2]))dK[1]\right )dK[2]=0,y(x)\right ] \]