Internal problem ID [1287]
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: 45.
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}+2 x +1\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (x +1\right ) y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, y^{\prime }\relax (0) = -2] \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+2*x+3*x^2)*diff(y(x),x$2)+(2-x^2)*diff(y(x),x)+(1+x)*y(x)=0,y(0) = 1, D(y)(0) = -2],y(x),type='series',x=0);
\[ y \relax (x ) = 1-2 x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {15}{8} x^{4}-\frac {71}{60} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 36
AsymptoticDSolveValue[{(1+2*x+3*x^2)*y''[x]+(2-x^2)*y'[x]+(1+x)*y[x]==0,{y[0]==1,y'[0]==-2}},y[x],{x,0,5}]
\[ y(x)\to -\frac {71 x^5}{60}+\frac {15 x^4}{8}-\frac {11 x^3}{6}+\frac {3 x^2}{2}-2 x+1 \]