20.25.19 problem 20

Internal problem ID [4024]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 20
Date solved : Monday, January 27, 2025 at 08:06:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 38

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

\[ y \left (x \right ) = \sqrt {x}\, \left (\left (x +\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{96} x^{4}+\frac {1}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 60

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

\[ y(x)\to c_2 \left (\sqrt {x} \left (\frac {x^5}{600}+\frac {x^4}{96}+\frac {x^3}{18}+\frac {x^2}{4}+x\right )+\sqrt {x} \log (x)\right )+c_1 \sqrt {x} \]