Internal problem ID [6197]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 3.
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 (x -3\right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 81
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(x-3)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{2} \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\frac {7}{720} x^{6}-\frac {1}{630} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}-\frac {463}{14400} x^{6}+\frac {971}{176400} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 164
AsymptoticDSolveValue[x^2*y''[x]+x*(x-3)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^7}{630}+\frac {7 x^6}{720}-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) x^2+c_2 \left (\left (\frac {971 x^7}{176400}-\frac {463 x^6}{14400}+\frac {187 x^5}{1200}-\frac {173 x^4}{288}+\frac {31 x^3}{18}-\frac {13 x^2}{4}+3 x\right ) x^2+\left (-\frac {x^7}{630}+\frac {7 x^6}{720}-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) x^2 \log (x)\right ) \]