67.2.41 problem Problem 18(f)

Internal problem ID [13998]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 18(f)
Date solved : Tuesday, January 28, 2025 at 08:25:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.123 (sec). Leaf size: 88

dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+2*(1-x)*y(x)=0,y(x), singsol=all)
 

\[ y = \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} \left (x c_{1} \left (x +2\right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )-x c_{2} \left (x +2\right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\left (\operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) c_{1} +\operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) c_{2} \right ) \left (-2+i \left (x +2\right ) \sqrt {7}+x^{2}+3 x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.170 (sec). Leaf size: 99

DSolve[x^2*D[y[x],{x,2}]+x^2*D[y[x],x]+2*(1-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {5}{2}+\frac {i \sqrt {7}}{2},1+i \sqrt {7},x\right )+c_2 L_{-\frac {1}{2} i \left (-5 i+\sqrt {7}\right )}^{i \sqrt {7}}(x)\right ) \exp \left (\int _1^x\frac {-2 K[1]+i \sqrt {7}+1}{2 K[1]}dK[1]\right ) \]