Internal problem ID [1337]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 48.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+8\right ) y^{\prime \prime }+7 x \left (x^{2}+2\right ) y^{\prime }-\left (-9 x^{2}+2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 35
Order:=6; dsolve(x^2*(8+x^2)*diff(y(x),x$2)+7*x*(2+x^2)*diff(y(x),x)-(2-9*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{4}} \left (1-\frac {13}{64} x^{2}+\frac {273}{8192} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x^{2}+\frac {2}{33} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 50
AsymptoticDSolveValue[x^2*(8+x^2)*y''[x]+7*x*(2+x^2)*y'[x]-(2-9*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {273 x^4}{8192}-\frac {13 x^2}{64}+1\right )+\frac {c_2 \left (\frac {2 x^4}{33}-\frac {x^2}{3}+1\right )}{x} \]