20.25.14 problem 15

Internal problem ID [4019]
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 : 15
Date solved : Monday, January 27, 2025 at 08:06:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 41

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

\[ y \left (x \right ) = \frac {c_{1} \left (1+3 x +\frac {21}{2} x^{2}-\frac {35}{2} x^{3}+\frac {35}{8} x^{4}-\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x^{2} \left (1-\frac {4}{7} x +\frac {4}{63} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 65

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

\[ y(x)\to c_1 \left (\frac {4 x^2}{63}-\frac {4 x}{7}+1\right ) x^2+\frac {c_2 \left (-\frac {7 x^5}{40}+\frac {35 x^4}{8}-\frac {35 x^3}{2}+\frac {21 x^2}{2}+3 x+1\right )}{\sqrt {x}} \]