Internal problem ID [5292]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 1(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+x -2\right )^{2} y^{\prime \prime }+3 \left (2+x \right ) y^{\prime }+\left (x -1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = -2\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 57
Order:=8; dsolve((x^2+x-2)^2*diff(y(x),x$2)+3*(x+2)*diff(y(x),x)+(x-1)*y(x)=0,y(x),type='series',x=-2);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {5}{9} \left (x +2\right )+\frac {23}{324} \left (x +2\right )^{2}+\frac {271}{43740} \left (x +2\right )^{3}+\frac {10517}{12597120} \left (x +2\right )^{4}+\frac {778801}{6235574400} \left (x +2\right )^{5}+\frac {16965493}{942818849280} \left (x +2\right )^{6}+\frac {899971067}{458981357990400} \left (x +2\right )^{7}+\mathrm {O}\left (\left (x +2\right )^{8}\right )\right )+c_{2} \left (x +2\right )^{\frac {4}{3}} \left (1-\frac {1}{21} \left (x +2\right )-\frac {11}{1260} \left (x +2\right )^{2}-\frac {53}{29484} \left (x +2\right )^{3}-\frac {11093}{28304640} \left (x +2\right )^{4}-\frac {709507}{8066822400} \left (x +2\right )^{5}-\frac {5797423}{290405606400} \left (x +2\right )^{6}-\frac {52991201}{11727918720000} \left (x +2\right )^{7}+\mathrm {O}\left (\left (x +2\right )^{8}\right )\right )}{\left (x +2\right )^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 148
AsymptoticDSolveValue[(x^2+x-2)^2*y''[x]+3*(x+2)*y'[x]+(x-1)*y[x]==0,y[x],{x,-2,7}]
\[ y(x)\to c_1 (x+2) \left (-\frac {52991201 (x+2)^7}{11727918720000}-\frac {5797423 (x+2)^6}{290405606400}-\frac {709507 (x+2)^5}{8066822400}-\frac {11093 (x+2)^4}{28304640}-\frac {53 (x+2)^3}{29484}-\frac {11 (x+2)^2}{1260}+\frac {1}{21} (-x-2)+1\right )+\frac {c_2 \left (\frac {899971067 (x+2)^7}{458981357990400}+\frac {16965493 (x+2)^6}{942818849280}+\frac {778801 (x+2)^5}{6235574400}+\frac {10517 (x+2)^4}{12597120}+\frac {271 (x+2)^3}{43740}+\frac {23}{324} (x+2)^2-\frac {5 (x+2)}{9}+1\right )}{\sqrt [3]{x+2}} \]