Internal problem ID [6214]
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: 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 (2+3 x \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: 0.027 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(2+3*x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\frac {27}{2240} x^{6}-\frac {81}{22400} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\frac {243}{20} x^{6}-\frac {729}{140} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 92
AsymptoticDSolveValue[x^2*y''[x]+x*(2+3*x)*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {81 x^4}{80}-\frac {81 x^3}{40}+\frac {27 x^2}{8}+\frac {1}{x^2}-\frac {9 x}{2}-\frac {3}{x}+\frac {9}{2}\right )+c_2 \left (\frac {27 x^7}{2240}-\frac {81 x^6}{2240}+\frac {27 x^5}{280}-\frac {9 x^4}{40}+\frac {9 x^3}{20}-\frac {3 x^2}{4}+x\right ) \]