Internal problem ID [1409]
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: 62.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (3 x +4\right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
Order:=6; dsolve(x^2*(4+3*x)*diff(y(x),x$2)-x*(4-3*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {3}{4} x +\frac {9}{16} x^{2}-\frac {27}{64} x^{3}+\frac {81}{256} x^{4}-\frac {243}{1024} x^{5}\right ) x \left (c_{2} \ln \relax (x )+c_{1}\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 84
AsymptoticDSolveValue[x^2*(4+3*x)*y''[x]-x*(4-3*x)*y'[x]+4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (-\frac {243 x^5}{1024}+\frac {81 x^4}{256}-\frac {27 x^3}{64}+\frac {9 x^2}{16}-\frac {3 x}{4}+1\right )+c_2 x \left (-\frac {243 x^5}{1024}+\frac {81 x^4}{256}-\frac {27 x^3}{64}+\frac {9 x^2}{16}-\frac {3 x}{4}+1\right ) \log (x) \]