Internal problem ID [1455]
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: 39.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1+2 x^{2}\right ) y^{\prime \prime }-x \left (x^{2}+3\right ) y^{\prime }-2 y x=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: 54
Order:=6; dsolve(x^2*(1+2*x^2)*diff(y(x),x$2)-x*(3+x^2)*diff(y(x),x)-2*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{4} \left (1+\frac {2}{5} x -\frac {8}{5} x^{2}-\frac {86}{105} x^{3}+\frac {445}{168} x^{4}+\frac {9571}{6300} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (24 x^{4}+\frac {48}{5} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-144+96 x -48 x^{2}+210 x^{4}+\frac {1812}{25} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 71
AsymptoticDSolveValue[x^2*(1+2*x^2)*y''[x]-x*(3+x^2)*y'[x]-2*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{12} \left (3 x^4+4 x^2-8 x+12\right )-\frac {1}{6} x^4 \log (x)\right )+c_2 \left (\frac {445 x^8}{168}-\frac {86 x^7}{105}-\frac {8 x^6}{5}+\frac {2 x^5}{5}+x^4\right ) \]