Internal problem ID [1300]
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: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+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: 44
Order:=6; dsolve(x^2*(5+x+10*x^2)*diff(y(x),x$2)+x*(4+3*x+48*x^2)*diff(y(x),x)+(x+36*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{5}} \left (1-\frac {6}{25} x -\frac {1217}{625} x^{2}+\frac {41972}{46875} x^{3}+\frac {1447799}{390625} x^{4}-\frac {375253322}{146484375} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{4} x -\frac {35}{18} x^{2}+\frac {11}{12} x^{3}+\frac {632}{171} x^{4}-\frac {2671}{1026} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 85
AsymptoticDSolveValue[x^2*(5+x+10*x^2)*y''[x]+x*(4+3*x+48*x^2)*y'[x]+(x+36*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [5]{x} \left (-\frac {375253322 x^5}{146484375}+\frac {1447799 x^4}{390625}+\frac {41972 x^3}{46875}-\frac {1217 x^2}{625}-\frac {6 x}{25}+1\right )+c_2 \left (-\frac {2671 x^5}{1026}+\frac {632 x^4}{171}+\frac {11 x^3}{12}-\frac {35 x^2}{18}-\frac {x}{4}+1\right ) \]