Internal problem ID [1356]
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: 4.
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^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (x +1\right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 81
Order:=8; dsolve(4*x^2*(1+x+x^2)*diff(y(x),x$2)+12*x^2*(1+x)*diff(y(x),x)+(1+3*x+3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-2 x +\frac {5}{2} x^{2}-2 x^{3}+\frac {5}{8} x^{4}+\frac {17}{20} x^{5}-\frac {121}{80} x^{6}+x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (x -\frac {9}{4} x^{2}+\frac {17}{6} x^{3}-\frac {205}{96} x^{4}+\frac {481}{1200} x^{5}+\frac {2109}{1600} x^{6}-\frac {1063}{560} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 156
AsymptoticDSolveValue[4*x^2*(1+x+x^2)*y''[x]+12*x^2*(1+x)*y'[x]+(1+3*x+3*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (x^7-\frac {121 x^6}{80}+\frac {17 x^5}{20}+\frac {5 x^4}{8}-2 x^3+\frac {5 x^2}{2}-2 x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {1063 x^7}{560}+\frac {2109 x^6}{1600}+\frac {481 x^5}{1200}-\frac {205 x^4}{96}+\frac {17 x^3}{6}-\frac {9 x^2}{4}+x\right )+\sqrt {x} \left (x^7-\frac {121 x^6}{80}+\frac {17 x^5}{20}+\frac {5 x^4}{8}-2 x^3+\frac {5 x^2}{2}-2 x+1\right ) \log (x)\right ) \]