Internal problem ID [1361]
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: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+x +4\right ) y^{\prime }+\left (3 x^{2}+5 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 81
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+2*x*(4+x+x^2)*diff(y(x),x)+(1+5*x+3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {1}{4} x^{2}+\frac {1}{18} x^{3}-\frac {37}{1152} x^{4}-\frac {17}{28800} x^{5}+\frac {593}{259200} x^{6}-\frac {1913}{12700800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {13}{16} x^{2}+\frac {1}{54} x^{3}+\frac {1103}{13824} x^{4}-\frac {19507}{1728000} x^{5}-\frac {98531}{20736000} x^{6}+\frac {982189}{889056000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 172
AsymptoticDSolveValue[4*x^2*y''[x]+2*x*(4+x+x^2)*y'[x]+(1+5*x+3*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {1913 x^7}{12700800}+\frac {593 x^6}{259200}-\frac {17 x^5}{28800}-\frac {37 x^4}{1152}+\frac {x^3}{18}+\frac {x^2}{4}-x+1\right )}{\sqrt {x}}+c_2 \left (\frac {\frac {982189 x^7}{889056000}-\frac {98531 x^6}{20736000}-\frac {19507 x^5}{1728000}+\frac {1103 x^4}{13824}+\frac {x^3}{54}-\frac {13 x^2}{16}+\frac {3 x}{2}}{\sqrt {x}}+\frac {\left (-\frac {1913 x^7}{12700800}+\frac {593 x^6}{259200}-\frac {17 x^5}{28800}-\frac {37 x^4}{1152}+\frac {x^3}{18}+\frac {x^2}{4}-x+1\right ) \log (x)}{\sqrt {x}}\right ) \]