Internal problem ID [1442]
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
III. Exercises 7.7. Page 389
Problem number: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 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: 49
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1+2*x^2)*diff(y(x),x)-(1-10*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {3}{2} x^{2}+x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (8 x^{2}-12 x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+2 x^{2}+4 x^{4}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 52
AsymptoticDSolveValue[x^2*y''[x]+x*(1+2*x^2)*y'[x]-(1-10*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (x^5-\frac {3 x^3}{2}+x\right )+c_1 \left (2 x \left (3 x^2-2\right ) \log (x)-\frac {5 x^4-x^2-1}{x}\right ) \]