Internal problem ID [5729]
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(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (1+x \right ) y^{\prime }-y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
Order:=8; dsolve(diff(y(x),x$2)-(x+1)*diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{30} x^{5}+\frac {1}{60} x^{6}+\frac {37}{5040} x^{7}\right ) y \relax (0)+\left (x +\frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{8} x^{5}+\frac {47}{720} x^{6}+\frac {19}{630} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 91
AsymptoticDSolveValue[y''[x]-(x+1)*y'[x]-x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {37 x^7}{5040}+\frac {x^6}{60}+\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}+1\right )+c_2 \left (\frac {19 x^7}{630}+\frac {47 x^6}{720}+\frac {x^5}{8}+\frac {x^4}{4}+\frac {x^3}{3}+\frac {x^2}{2}+x\right ) \]