Internal problem ID [1252]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (2+x \right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+y=0} \end {gather*} With initial conditions \begin {align*} [y \left (-1\right ) = -2, y^{\prime }\left (-1\right ) = 3] \end {align*}
With the expansion point for the power series method at \(x = -1\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 20
Order:=6; dsolve([(2+x)*diff(y(x),x$2)+(2+x)*diff(y(x),x)+y(x)=0,y(-1) = -2, D(y)(-1) = 3],y(x),type='series',x=-1);
\[ y \relax (x ) = -2+3 \left (x +1\right )-\frac {1}{2} \left (x +1\right )^{2}-\frac {2}{3} \left (x +1\right )^{3}+\frac {5}{8} \left (x +1\right )^{4}-\frac {11}{30} \left (x +1\right )^{5}+\mathrm {O}\left (\left (x +1\right )^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 46
AsymptoticDSolveValue[{(2+x)*y''[x]+(2+x)*y'[x]+y[x]==0,{y[-1]==-2,y'[-1]==3}},y[x],{x,-1,5}]
\[ y(x)\to -\frac {11}{30} (x+1)^5+\frac {5}{8} (x+1)^4-\frac {2}{3} (x+1)^3-\frac {1}{2} (x+1)^2+3 (x+1)-2 \]