Internal problem ID [1425]
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: 9.
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 (3+8 x \right ) y^{\prime }-\left (5-49 x \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: 65
Order:=6; dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x*(3+8*x)*diff(y(x),x)-(5-49*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{3} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\left (-36\right ) x^{3}+144 x^{4}-360 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (12+6 x +12 x^{2}-240 x^{3}+852 x^{4}-2022 x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{\frac {5}{2}}} \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 86
AsymptoticDSolveValue[4*x^2*(1+x)*y''[x]+4*x*(3+8*x)*y'[x]-(5-49*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (35 x^{9/2}-20 x^{7/2}+10 x^{5/2}-4 x^{3/2}+\sqrt {x}\right )+c_1 \left (\frac {62 x^4-20 x^3+2 x^2+x+2}{2 x^{5/2}}+3 \sqrt {x} (4 x-1) \log (x)\right ) \]