Internal problem ID [10914]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 123
dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+2*(1-x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} \left (\left (x^{2}+2 x \right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\left (-2+i \left (x +2\right ) \sqrt {7}+x^{2}+3 x \right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right )\right )+c_{2} \left (\left (-x^{2}-2 x \right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\left (-2+i \left (x +2\right ) \sqrt {7}+x^{2}+3 x \right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right )\right ) \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 89
DSolve[x^2*y''[x]+x^2*y'[x]+2*(1-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} x^{\frac {1}{2}+\frac {i \sqrt {7}}{2}} \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 ) \\ \end{align*}