83.5.13 problem 13

Internal problem ID [19108]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (D) at page 16
Problem number : 13
Date solved : Tuesday, January 28, 2025 at 12:57:06 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 18

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

\[ y \left (x \right ) = \tan \left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-x -1}\right )+x +1\right ) \]

✓ Solution by Mathematica

Time used: 60.117 (sec). Leaf size: 21

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

\[ y(x)\to \tan \left (W\left (c_1 \left (-e^{-x-1}\right )\right )+x+1\right ) \]