Internal problem ID [1437]
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: 21.
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 (4 x +1\right ) y^{\prime }-\left (49+27 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
Order:=6; dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x*(1+4*x)*diff(y(x),x)-(49+27*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{7} \left (1-2 x +3 x^{2}-4 x^{3}+5 x^{4}-6 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (3628800-3024000 x +2419200 x^{2}-1814400 x^{3}+1209600 x^{4}-604800 x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {7}{2}}} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 86
AsymptoticDSolveValue[4*x^2*(1+x)*y''[x]+4*x*(1+4*x)*y'[x]-(49+27*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {2}{3 x^{3/2}}-\frac {5}{6 x^{5/2}}+\frac {1}{x^{7/2}}+\frac {\sqrt {x}}{3}-\frac {1}{2 \sqrt {x}}\right )+c_2 \left (5 x^{15/2}-4 x^{13/2}+3 x^{11/2}-2 x^{9/2}+x^{7/2}\right ) \]