Internal problem ID [5725]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert.
(A) Drill Exercises . Page 194
Problem number: 1(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime }+y-x^{3}+x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 63
Order:=8; dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=x^3-x,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) D\relax (y )\relax (0)-\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{20}-\frac {7 x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 105
AsymptoticDSolveValue[y''[x]+y'[x]+y[x]==x^3-x,y[x],{x,0,7}]
\[ y(x)\to \frac {x^7}{5040}-\frac {7 x^6}{720}+\frac {x^5}{20}+\frac {x^4}{24}-\frac {x^3}{6}+c_2 \left (\frac {x^7}{5040}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]