62.38.10 problem Ex 10

Internal problem ID [13021]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 10
Date solved : Tuesday, January 28, 2025 at 04:49:26 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 3.578 (sec). Leaf size: 84

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

\[ y(x)\to \int _1^x\exp \left (\int _1^{K[3]}\frac {1}{K[1]-K[1]^3}dK[1]\right ) \left (c_1+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {1}{K[1]-K[1]^3}dK[1]\right ) K[2]^2}{K[2]^2-1}dK[2]\right )dK[3]+c_2 \]