81.3.4 problem 4

Internal problem ID [18622]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 08:28:44 PM
CAS classification : [_linear]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+x^{2} y&=x^{3}-x^{2} \arctan \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 55

dsolve((1+x^2)*diff(y(x),x)+x^2*y(x)=x^3-x^2*arctan(x),y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {{\mathrm e}^{\arctan \left (x \right )-x} \left (\left (i \ln \left (-i x +1\right )-i \ln \left (i x +1\right )-2 x +2\right ) {\mathrm e}^{x} \left (i x +1\right )^{\frac {i}{2}} \left (-i x +1\right )^{-\frac {i}{2}}-2 c_{1} \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.129 (sec). Leaf size: 23

DSolve[(1+x^2)*D[y[x],x]+x^2*y[x]==x^3-x^2*ArcTan[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\arctan (x)+c_1 e^{\arctan (x)-x}+x-1 \]