7.15.32 problem 32

Internal problem ID [488]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 32
Date solved : Monday, January 27, 2025 at 02:54:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x \left (1+x \right ) 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: 0.007 (sec). Leaf size: 37

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

\[ y = \frac {c_1 \left (1-\frac {5}{2} x -\frac {15}{8} x^{2}-\frac {5}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{8960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{5} x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 58

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

\[ y(x)\to \frac {c_2 \left (-\frac {x^5}{8960}+\frac {x^4}{384}-\frac {5 x^3}{48}-\frac {15 x^2}{8}-\frac {5 x}{2}+1\right )}{\sqrt {x}}+c_1 \left (\frac {x}{5}+1\right ) x \]