Internal problem ID [1271]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN
ORDINARY POINT II. Exercises 7.3. Page 338
Problem number: 31(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}+4 x +4\right ) y^{\prime \prime }+\left (8+4 x \right ) y^{\prime }+2 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: 54
Order:=6; dsolve((4+4*x+x^2)*diff(y(x),x$2)+(8+4*x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{16} x^{4}+\frac {1}{8} x^{5}\right ) y \relax (0)+\left (x -x^{2}+\frac {3}{4} x^{3}-\frac {1}{2} x^{4}+\frac {5}{16} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 68
AsymptoticDSolveValue[(4+4*x+x^2)*y''[x]+(8+4*x)*y'[x]+2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^5}{8}-\frac {3 x^4}{16}+\frac {x^3}{4}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {5 x^5}{16}-\frac {x^4}{2}+\frac {3 x^3}{4}-x^2+x\right ) \]