Internal problem ID [11903]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{3}+x^{2}\right ) y^{\prime \prime }+\left (x^{2}-2 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 1227
Order:=6; dsolve((x^3+x^2)*diff(y(x),x$2)+(x^2-2*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = x^{\frac {3}{2}} \left (c_{2} x^{\frac {i \sqrt {7}}{2}} \left (1+\frac {3 \sqrt {7}-i}{-2 \sqrt {7}+2 i} x +\frac {-4 \sqrt {7}-12 i}{\left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right )} x^{2}+\frac {224}{3} \frac {1}{\left (\sqrt {7}-2 i\right ) \left (-\sqrt {7}+i\right ) \left (3+i \sqrt {7}\right )} x^{3}+\frac {84 \sqrt {7}-\frac {1036 i}{3}}{\left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right ) \left (3+i \sqrt {7}\right ) \left (4+i \sqrt {7}\right )} x^{4}+\frac {\frac {2576 i \sqrt {7}}{3}+\frac {6608}{5}}{\left (-4 i+\sqrt {7}\right ) \left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right ) \left (3+i \sqrt {7}\right ) \left (i \sqrt {7}+5\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {7}}{2}} \left (1+\frac {-3 \sqrt {7}-i}{2 \sqrt {7}+2 i} x +\frac {12+4 i \sqrt {7}}{5+3 i \sqrt {7}} x^{2}+\frac {224}{3} \frac {1}{\left (i \sqrt {7}-2\right ) \left (\sqrt {7}+3 i\right ) \left (\sqrt {7}+i\right )} x^{3}+\frac {63 i \sqrt {7}-259}{15 i \sqrt {7}-129} x^{4}+\frac {-1239 i-805 \sqrt {7}}{675 i+255 \sqrt {7}} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 5834
AsymptoticDSolveValue[(x^3+x^2)*y''[x]+(x^2-2*x)*y'[x]+4*y[x]==0,y[x],{x,0,5}]
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