75.20.7 problem 642

Internal problem ID [17141]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 642
Date solved : Tuesday, January 28, 2025 at 09:54:07 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 18

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

\[ y = \sqrt {x^{2}+1}\, c_{2} +c_{1} x -1 \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 26

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

\[ y(x)\to c_1 \sqrt {x^2+1}+i c_2 x-1 \]