Internal problem ID [4778]
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: 7.3.103.
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 }+\left (x -\frac {3}{4}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 65
Order:=6; dsolve(x^2*diff(y(x),x$2)+(x-3/4)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{3} x +\frac {1}{24} x^{2}-\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}-\frac {1}{302400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\left (x^{2}-\frac {1}{3} x^{3}+\frac {1}{24} x^{4}-\frac {1}{360} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x )+\left (-2-2 x +\frac {4}{9} x^{3}-\frac {25}{288} x^{4}+\frac {157}{21600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 101
AsymptoticDSolveValue[x^2*y''[x]+(x-3/4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{11/2}}{8640}-\frac {x^{9/2}}{360}+\frac {x^{7/2}}{24}-\frac {x^{5/2}}{3}+x^{3/2}\right )+c_1 \left (\frac {31 x^4-176 x^3+144 x^2+576 x+576}{576 \sqrt {x}}-\frac {1}{48} x^{3/2} \left (x^2-8 x+24\right ) \log (x)\right ) \]