Internal problem ID [442]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number: problem 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}+1\right ) y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 1, 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([diff(y(x),x$2)+x*diff(y(x),x)+(2*x^2+1)*y(x)=0,y(0) = 1, D(y)(0) = -1],y(x),type='series',x=0);
\[ y \relax (x ) = 1-x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 49
AsymptoticDSolveValue[{(x^2+1)*y''[x]+2*x*y'[x]+2*x*y[x]==0,{}},y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^5}{5}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^5}{5}-\frac {x^4}{6}-\frac {x^3}{3}+x\right ) \]