Internal problem ID [1391]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 39.
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}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }+\left (12 x^{2}+1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(3+8*x^2)*diff(y(x),x)+(1+12*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 71
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]+x*(3+8*x^2)*y'[x]+(1+12*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right )}{x}+c_2 \left (\frac {\frac {x^2}{4}-\frac {13 x^4}{32}}{x}+\frac {\left (\frac {15 x^4}{8}-\frac {3 x^2}{2}+1\right ) \log (x)}{x}\right ) \]