Internal problem ID [1445]
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: 29.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (x^{2}+1\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: 0.012 (sec). Leaf size: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1+x^2)*diff(y(x),x)-(1-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (2 x^{2}-x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+x^{2}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 51
AsymptoticDSolveValue[x^2*y''[x]+x*(1+x^2)*y'[x]-(1-3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{8}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {1}{2} x \left (x^2-2\right ) \log (x)-\frac {x^4-4}{4 x}\right ) \]