Internal problem ID [14011]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page
739
Problem number: 36.2 (g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x +\left (1-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 69
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+8*x*diff(y(x),x)+(1-4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 124
AsymptoticDSolveValue[4*x^2*y''[x]+8*x*y'[x]+(1-4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right )}{\sqrt {x}}+c_2 \left (\frac {-\frac {137 x^5}{432000}-\frac {25 x^4}{3456}-\frac {11 x^3}{108}-\frac {3 x^2}{4}-2 x}{\sqrt {x}}+\frac {\left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right ) \log (x)}{\sqrt {x}}\right ) \]