Internal problem ID [6239]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (3-x \right ) y^{\prime }-5 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 76
Order:=8; dsolve(x*diff(y(x),x$2)+(3-x)*diff(y(x),x)-5*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1+\frac {5}{3} x +\frac {5}{4} x^{2}+\frac {7}{12} x^{3}+\frac {7}{36} x^{4}+\frac {1}{20} x^{5}+\frac {1}{96} x^{6}+\frac {11}{6048} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \relax (x ) \left (12 x^{2}+20 x^{3}+15 x^{4}+7 x^{5}+\frac {7}{3} x^{6}+\frac {3}{5} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-2+6 x +7 x^{2}-11 x^{3}-17 x^{4}-\frac {32}{3} x^{5}-\frac {305}{72} x^{6}-\frac {737}{600} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.16 (sec). Leaf size: 116
AsymptoticDSolveValue[x*y''[x]+(3-x)*y'[x]-5*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^6}{96}+\frac {x^5}{20}+\frac {7 x^4}{36}+\frac {7 x^3}{12}+\frac {5 x^2}{4}+\frac {5 x}{3}+1\right )+c_1 \left (\frac {389 x^6+1020 x^5+1764 x^4+1512 x^3-72 x^2-432 x+144}{144 x^2}-\frac {1}{6} \left (7 x^4+21 x^3+45 x^2+60 x+36\right ) \log (x)\right ) \]