Internal problem ID [1292]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: Example 7.5.1 page 353.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 45
Order:=6; dsolve(2*x^2*(1+x+x^2)*diff(y(x),x$2)+x*(9+11*x+11*x^2)*diff(y(x),x)+(6+10*x+7*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {1}{8} x^{4}+\frac {1}{30} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1} \sqrt {x}+\left (1-\frac {1}{3} x +\frac {2}{5} x^{2}-\frac {5}{21} x^{3}+\frac {7}{135} x^{4}+\frac {76}{1155} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} x}{x^{\frac {5}{2}}} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 83
AsymptoticDSolveValue[2*x^2*(1+x+x^2)*y''[x]+x*(9+11*x+11*x^2)*y'[x]+(6+10*x+7*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_2 \left (\frac {x^5}{30}+\frac {x^4}{8}-\frac {x^3}{3}+\frac {x^2}{2}+1\right )}{x^2}+\frac {c_1 \left (\frac {76 x^5}{1155}+\frac {7 x^4}{135}-\frac {5 x^3}{21}+\frac {2 x^2}{5}-\frac {x}{3}+1\right )}{x^{3/2}} \]