62.38.2 problem Ex 2

Internal problem ID [13013]
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 2
Date solved : Tuesday, January 28, 2025 at 04:49:14 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 59

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

\[ y = -\int -\frac {-2 \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {x -1}\, \sqrt {x +1}+c_{1} \left (x^{2}-1\right )}{\left (x -1\right )^{{3}/{2}} \left (x +1\right )^{{3}/{2}}}d x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 31

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

\[ y(x)\to c_2-\frac {1}{4} \left (-2 \log \left (\sqrt {x^2-1}+x\right )+c_1\right ){}^2 \]