62.12.15 problem Ex 16

Internal problem ID [12861]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 16
Date solved : Tuesday, January 28, 2025 at 04:28:54 AM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

Time used: 0.193 (sec). Leaf size: 69

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

\[ y(x)\to \exp \left (\int _1^x-\frac {1}{K[1]^2+1}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {1}{K[1]^2+1}dK[1]\right ) \arctan (K[2])}{K[2]^2+1}dK[2]+c_1\right ) \]