Internal problem ID [4883]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG,
EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power
Series Method: Frobenius Method page 186
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x -2\right )^{2} y^{\prime \prime }+\left (2+x \right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 44
Order:=6; dsolve((x-2)^2*diff(y(x),x$2)+(x+2)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{8} x^{2}+\frac {1}{48} x^{3}-\frac {1}{480} x^{5}\right ) y \relax (0)+\left (x -\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{240} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[(x-2)^2*y''[x]+(x+2)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {x^5}{480}+\frac {x^3}{48}+\frac {x^2}{8}+1\right )+c_2 \left (\frac {x^5}{240}-\frac {x^3}{24}-\frac {x^2}{4}+x\right ) \]