83.39.3 problem 3

Internal problem ID [19471]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (D) at page 135
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 01:45:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 2.968 (sec). Leaf size: 119

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

\[ y \left (x \right ) = \sqrt {x}\, \operatorname {HeunD}\left (4 \sqrt {2}, -1-4 \sqrt {2}, 8 \sqrt {2}, -4 \sqrt {2}+1, \frac {\sqrt {2}\, x -1}{\sqrt {2}\, x +1}\right ) {\mathrm e}^{-\frac {\left (x -1\right ) \left (x +1\right )}{x}} c_{2} +\sqrt {x}\, {\mathrm e}^{x} \operatorname {HeunD}\left (-4 \sqrt {2}, -1-4 \sqrt {2}, 8 \sqrt {2}, -4 \sqrt {2}+1, \frac {\sqrt {2}\, x -1}{\sqrt {2}\, x +1}\right ) c_{1} -\frac {{\mathrm e}^{-x}}{2} \]

✓ Solution by Mathematica

Time used: 0.300 (sec). Leaf size: 81

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

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