20.24.13 problem Problem 13

Internal problem ID [3998]
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 11.2. page 739
Problem number : Problem 13
Date solved : Monday, January 27, 2025 at 08:05:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+4 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 54

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

\[ y \left (x \right ) = \left (1-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}-\frac {2}{15} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {2}{3} x^{3}-\frac {2}{3} x^{4}+\frac {7}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 61

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

\[ y(x)\to c_1 \left (-\frac {2 x^5}{15}+\frac {x^4}{3}-\frac {2 x^3}{3}+1\right )+c_2 \left (\frac {7 x^5}{15}-\frac {2 x^4}{3}+\frac {2 x^3}{3}-x^2+x\right ) \]