11.5 problem 15

Internal problem ID [1194]

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: 15.
ODE order: 2.
ODE degree: 1.

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

Solve \begin {gather*} \boxed {\left (3 x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.004 (sec). Leaf size: 32

Order:=6; 
dsolve((1+3*x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

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

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

AsymptoticDSolveValue[(1+3*x^2)*y''[x]-2*x*y'[x]+4*y[x]==0,y[x],{x,0,5}]
 

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