12.11 problem 11

Internal problem ID [1215]

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: 11.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]

Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 2, y^{\prime }\relax (0) = -1] \end {align*}

With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.0 (sec). Leaf size: 20

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

\[ y \relax (x ) = 2-x -x^{2}+\frac {1}{3} x^{3}+\frac {5}{12} x^{4}-\frac {1}{6} x^{5}+\mathrm {O}\left (x^{6}\right ) \]

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {x^5}{6}+\frac {5 x^4}{12}+\frac {x^3}{3}-x^2-x+2 \]