Internal problem ID [4530]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {z^{\prime \prime }+x z^{\prime }+z-x^{2}-2 x -1=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 42
Order:=6; dsolve(diff(z(x),x$2)+x*diff(z(x),x)+z(x)=x^2+2*x+1,z(x),type='series',x=0);
\[ z \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) z \relax (0)+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) D\relax (z )\relax (0)+\frac {x^{2}}{2}+\frac {x^{3}}{3}-\frac {x^{4}}{24}-\frac {x^{5}}{15}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 70
AsymptoticDSolveValue[z''[x]+x*z'[x]+z[x]==x^2+2*x+1,z[x],{x,0,5}]
\[ z(x)\to -\frac {x^5}{15}-\frac {x^4}{24}+\frac {x^3}{3}+\frac {x^2}{2}+c_2 \left (\frac {x^5}{15}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]