Internal problem ID [11923]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\frac {3 y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 65
Order:=6; dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)-3/4*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{2} x +\frac {5}{32} x^{2}-\frac {7}{192} x^{3}+\frac {7}{1024} x^{4}-\frac {11}{10240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{4} x^{2}+\frac {1}{8} x^{3}-\frac {5}{128} x^{4}+\frac {7}{768} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+x -\frac {1}{12} x^{3}+\frac {61}{1536} x^{4}-\frac {59}{5120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 103
AsymptoticDSolveValue[x^2*y''[x]+x^2*y'[x]-3/4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {7 x^{11/2}}{1024}-\frac {7 x^{9/2}}{192}+\frac {5 x^{7/2}}{32}-\frac {x^{5/2}}{2}+x^{3/2}\right )+c_1 \left (\frac {1}{256} x^{3/2} \left (5 x^2-16 x+32\right ) \log (x)-\frac {91 x^4-224 x^3+192 x^2+1536 x-3072}{3072 \sqrt {x}}\right ) \]