47.5.10 problem 10

Internal problem ID [7499]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 10
Date solved : Monday, January 27, 2025 at 03:02:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 53

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

\[ y = \frac {2 \left (x^{2}+7\right ) \sqrt {x +\sqrt {x^{2}-1}}+15 c_{1} x +15 c_{1} \sqrt {x^{2}-1}+15 c_{2}}{15 \sqrt {x +\sqrt {x^{2}-1}}} \]

✓ Solution by Mathematica

Time used: 7.781 (sec). Leaf size: 228

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

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