Internal problem ID [1419]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }-\left (3 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 51
Order:=6; dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x*(1+2*x)*diff(y(x),x)-(1+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x \left (1+\mathrm {O}\left (x^{6}\right )\right )+\ln \relax (x ) \left (x +\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (1-x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{3} x^{4}+\frac {1}{4} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 53
AsymptoticDSolveValue[4*x^2*(1+x)*y''[x]+4*x*(1+2*x)*y'[x]-(1+3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\sqrt {x} \log (x)-\frac {2 x^4-3 x^3+6 x^2+6 x-6}{6 \sqrt {x}}\right )+c_2 \sqrt {x} \]