Internal problem ID [4762]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page
300
Problem number: 7.3.3.
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 y^{\prime }+\left (x +1\right ) 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: 69
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 90
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {1}{12480}+\frac {i}{2496}\right ) c_2 x^{-i} \left (i x^4-(8+16 i) x^3+(168+96 i) x^2-(1056-288 i) x+(480-2400 i)\right )-\left (\frac {1}{2496}+\frac {i}{12480}\right ) c_1 x^i \left (x^4-(16+8 i) x^3+(96+168 i) x^2+(288-1056 i) x-(2400-480 i)\right ) \]