Internal problem ID [6215]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a
Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 50
Order:=8; dsolve(x*diff(y(x),x$2)-(3+x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{4} \left (1+\frac {2}{5} x +\frac {1}{10} x^{2}+\frac {2}{105} x^{3}+\frac {1}{336} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{21600} x^{6}+\frac {1}{207900} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (-144-96 x -24 x^{2}+2 x^{4}+\frac {4}{5} x^{5}+\frac {1}{5} x^{6}+\frac {4}{105} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.153 (sec). Leaf size: 91
AsymptoticDSolveValue[x*y''[x]-(3+x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^6}{720}-\frac {x^5}{180}-\frac {x^4}{72}+\frac {x^2}{6}+\frac {2 x}{3}+1\right )+c_2 \left (\frac {x^{10}}{21600}+\frac {x^9}{2520}+\frac {x^8}{336}+\frac {2 x^7}{105}+\frac {x^6}{10}+\frac {2 x^5}{5}+x^4\right ) \]