12.11.1 problem 11

Internal problem ID [1840]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 11
Date solved : Monday, January 27, 2025 at 05:37:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {3}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{8} x^{4}-\frac {9}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 63

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

\[ y(x)\to c_2 \left (\frac {x^5}{40}+\frac {x^4}{12}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {9 x^5}{160}+\frac {x^4}{8}+\frac {x^3}{8}-\frac {3 x^2}{4}+1\right ) \]