64.14.5 problem 5

Internal problem ID [13564]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 05:53:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (3 x +2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

Order:=6; 
dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(3*x+2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-x^{2}-\frac {1}{2} x^{3}+\frac {1}{3} x^{4}+\frac {11}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 61

AsymptoticDSolveValue[D[y[x],{x,2}]+x*D[y[x],x]+(3*x+2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{8}-\frac {x^4}{4}-\frac {x^3}{2}+x\right )+c_1 \left (\frac {11 x^5}{40}+\frac {x^4}{3}-\frac {x^3}{2}-x^2+1\right ) \]