81.2.10 problem 10

Internal problem ID [18618]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 12:04:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} v^{\prime \prime }+\frac {2 x v^{\prime }}{x^{2}+1}+\frac {v}{\left (x^{2}+1\right )^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 17

dsolve(diff(v(x),x$2)+2*x/(1+x^2)*diff(v(x),x)+v(x)/(1+x^2)^2=0,v(x), singsol=all)
 

\[ v \left (x \right ) = \frac {c_{1} x +c_{2}}{\sqrt {x^{2}+1}} \]

✓ Solution by Mathematica

Time used: 2.036 (sec). Leaf size: 22

DSolve[D[v[x],{x,2}]+2*x/(1+x^2)*D[v[x],x]+v[x]/(1+x^2)^2==0,v[x],x,IncludeSingularSolutions -> True]
 

\[ v(x)\to \frac {c_2 x+c_1}{\sqrt {x^2+1}} \]