Internal problem ID [1450]
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: 34.
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 (-3 x^{2}+1\right ) y^{\prime }-4 \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: 0.011 (sec). Leaf size: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-3*x^2)*diff(y(x),x)-4*(1-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1-\frac {1}{2} x^{2}+\mathrm {O}\left (x^{6}\right )\right )+\ln \relax (x ) \left (1944 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (-144-648 x^{2}-810 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 50
AsymptoticDSolveValue[x^2*y''[x]+x*(1-3*x^2)*y'[x]-4*(1-3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (x^2-\frac {x^4}{2}\right )+c_1 \left (\frac {18 x^4+9 x^2+2}{2 x^2}-\frac {27}{2} x^2 \log (x)\right ) \]