83.30.8 problem 8

Internal problem ID [19387]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (D) at page 109
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 01:36:32 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 61

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

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

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 45

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

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