83.27.9 problem 9

Internal problem ID [19355]
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 (A) at page 104
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:35:16 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.306 (sec). Leaf size: 214

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

\[ y(x)\to \cosh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right ) \int _1^x\frac {2 K[1] \sinh \left (\frac {\arcsin (K[1]) \sqrt {1-K[1]^2}}{\sqrt {K[1]^2-1}}\right )}{\sqrt {K[1]^2-1}}dK[1]+i \sinh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right ) \int _1^x\frac {2 i \cosh \left (\frac {\arcsin (K[2]) \sqrt {1-K[2]^2}}{\sqrt {K[2]^2-1}}\right ) K[2]}{\sqrt {K[2]^2-1}}dK[2]+c_1 \cosh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right )+i c_2 \sinh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right ) \]