Internal problem ID [1404]
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: 52.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 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(4*x^2*diff(y(x),x$2)+2*x*(4-x^2)*diff(y(x),x)+(1+7*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {5}{8} x^{2}-\frac {9}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 77
AsymptoticDSolveValue[4*x^2*y''[x]+2*x*(4-x^2)*y'[x]+(1+7*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (\frac {x^4}{32}-\frac {x^2}{2}+1\right )}{\sqrt {x}}+c_2 \left (\frac {\frac {5 x^2}{8}-\frac {9 x^4}{128}}{\sqrt {x}}+\frac {\left (\frac {x^4}{32}-\frac {x^2}{2}+1\right ) \log (x)}{\sqrt {x}}\right ) \]