73.26.13 problem 36.6 (a)

Internal problem ID [15753]
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 : 36.6 (a)
Date solved : Tuesday, January 28, 2025 at 08:06:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 48

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+(1-4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 124

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+(1-4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {137 x^5}{432000}-\frac {25 x^4}{3456}-\frac {11 x^3}{108}-\frac {3 x^2}{4}-2 x\right )+\sqrt {x} \left (\frac {x^5}{14400}+\frac {x^4}{576}+\frac {x^3}{36}+\frac {x^2}{4}+x+1\right ) \log (x)\right ) \]