13.11 problem 13

Internal problem ID [721]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number: 13.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {2 y^{\prime \prime }+y^{\prime } x +3 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.0 (sec). Leaf size: 34

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

\[ y \relax (x ) = \left (1-\frac {3}{4} x^{2}+\frac {5}{32} x^{4}\right ) y \relax (0)+\left (x -\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]

Solution by Mathematica

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

AsymptoticDSolveValue[2*y''[x]+x*y'[x]+3*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{20}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {5 x^4}{32}-\frac {3 x^2}{4}+1\right ) \]