47.4.3 problem 51

Internal problem ID [7480]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 51
Date solved : Monday, January 27, 2025 at 03:01:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 15

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

\[ y = c_{1} \sin \left (\operatorname {arcsinh}\left (x \right )\right )+c_{2} \cos \left (\operatorname {arcsinh}\left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \cos (\text {arcsinh}(x))+c_2 \sin (\text {arcsinh}(x)) \]