Internal problem ID [5737]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert.
(A) Drill Exercises . Page 194
Problem number: 2(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 85
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{10}-\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}-\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}-\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}-\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}+\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}+\frac {1009 i}{575452800}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{i} \left (1+\left (\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{10}+\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}+\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}+\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}+\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}-\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}-\frac {1009 i}{575452800}\right ) x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*y''[x]+x*(1-x)*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \left (\frac {59}{10857600}-\frac {17 i}{10857600}\right ) c_2 x^{-i} \left ((14+5 i) x^6+(108+24 i) x^5+(720+60 i) x^4+(4080-240 i) x^3+(19440-3600 i) x^2+(77760-14400 i) x+(169920+48960 i)\right )+\left (\frac {59}{10857600}+\frac {17 i}{10857600}\right ) c_1 x^i \left ((14-5 i) x^6+(108-24 i) x^5+(720-60 i) x^4+(4080+240 i) x^3+(19440+3600 i) x^2+(77760+14400 i) x+(169920-48960 i)\right ) \]