Internal problem ID [1390]
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: 38.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (7 x^{2}+3\right ) y^{\prime }+\left (-3 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: 0.009 (sec). Leaf size: 51
Order:=6; dsolve(x^2*(1+2*x^2)*diff(y(x),x$2)+x*(3+7*x^2)*diff(y(x),x)+(1-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {3}{2} x^{2}-\frac {3}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {7}{4} x^{2}-\frac {7}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 71
AsymptoticDSolveValue[x^2*(1+2*x^2)*y''[x]+x*(3+7*x^2)*y'[x]+(1-3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (-\frac {3 x^4}{8}+\frac {3 x^2}{2}+1\right )}{x}+c_2 \left (\frac {-\frac {7 x^4}{32}-\frac {7 x^2}{4}}{x}+\frac {\left (-\frac {3 x^4}{8}+\frac {3 x^2}{2}+1\right ) \log (x)}{x}\right ) \]