Internal problem ID [1382]
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: 30.
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 y^{\prime } x^{3}+\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: 0.0 (sec). Leaf size: 51
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+2*x^3*diff(y(x),x)+(1+3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{8} x^{2}-\frac {3}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 77
AsymptoticDSolveValue[4*x^2*y''[x]+2*x^3*y'[x]+(1+3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{32}-\frac {x^2}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {x^2}{8}-\frac {3 x^4}{128}\right )+\sqrt {x} \left (\frac {x^4}{32}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]