73.17.44 problem 44

Internal problem ID [15578]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 25. Review exercises for part III. page 447
Problem number : 44
Date solved : Tuesday, January 28, 2025 at 08:02:39 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-y&=\frac {1}{x^{2}+1} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 63

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

\[ y(x)\to \frac {x^2 \int _1^x\frac {1}{2 \left (K[2]^4+K[2]^2\right )}dK[2]+\int _1^x-\frac {1}{2 \left (K[1]^2+1\right )}dK[1]+c_2 x^2+c_1}{x} \]