83.38.7 problem 7

Internal problem ID [19466]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 01:45:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 24

dsolve((1+x^2)^2*diff(y(x),x$2)+2*x*(1+x^2)*diff(y(x),x)+4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 22

DSolve[(1+x^2)^2*D[y[x],{x,2}]+(2*x)*(1+x^2)*D[y[x],x]+4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cos (2 \arctan (x))+c_2 \sin (2 \arctan (x)) \]