Internal problem ID [1301]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {8 x^{2} y^{\prime \prime }-2 x \left (-x^{2}-4 x +3\right ) y^{\prime }+\left (x^{2}+6 x +3\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: 47
Order:=6; dsolve(8*x^2*diff(y(x),x$2)-2*x*(3-4*x-x^2)*diff(y(x),x)+(3+6*x+x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{4}} \left (1+4 x -\frac {131}{24} x^{2}+\frac {39}{14} x^{3}-\frac {19865}{29568} x^{4}+\frac {4421}{110880} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{2}} \left (1-x +\frac {11}{26} x^{2}-\frac {109}{1326} x^{3}+\frac {5}{12376} x^{4}+\frac {229}{71400} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 86
AsymptoticDSolveValue[8*x^2*y''[x]-2*x*(3-4*x-x^2)*y'[x]+(3+6*x+x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {4421 x^5}{110880}-\frac {19865 x^4}{29568}+\frac {39 x^3}{14}-\frac {131 x^2}{24}+4 x+1\right ) \sqrt [4]{x}+c_1 \left (\frac {229 x^5}{71400}+\frac {5 x^4}{12376}-\frac {109 x^3}{1326}+\frac {11 x^2}{26}-x+1\right ) x^{3/2} \]