50.22.3 problem 1(c)

Internal problem ID [8146]
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)
Date solved : Monday, January 27, 2025 at 03:44:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=x^{3}-x \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 84

Order:=8; 
dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)=x^3-x,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{5040} x^{7}\right ) y^{\prime }\left (0\right )-\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{20}-\frac {7 x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 105

AsymptoticDSolveValue[D[y[x],{x,2}]+D[y[x],x]+y[x]==x^3-x,y[x],{x,0,"8"-1}]
 

\[ y(x)\to \frac {x^7}{5040}-\frac {7 x^6}{720}+\frac {x^5}{20}+\frac {x^4}{24}-\frac {x^3}{6}+c_2 \left (\frac {x^7}{5040}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]