73.26.7 problem 36.2 (g)

Internal problem ID [15747]
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)
Date solved : Tuesday, January 28, 2025 at 08:06:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x y^{\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: 0.048 (sec). Leaf size: 48

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);
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 124

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+8*x*D[y[x],x]+(1-4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

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