12.12.10 problem 10

Internal problem ID [1864]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 10
Date solved : Monday, January 27, 2025 at 05:37:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \left (1-\frac {3}{2} x^{2}+\frac {7}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {3}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {3 x^5}{8}-\frac {5 x^3}{6}+x\right )+c_1 \left (\frac {7 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]