Internal problem ID [5676]
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. Section 4.3. Second-Order Linear
Equations: Ordinary Points. Page 169
Problem number: 1(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 x y^{\prime }-y-x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 50
Order:=8; dsolve(diff(y(x),x$2)+2*x*diff(y(x),x)-y(x)=x,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{8} x^{4}+\frac {7}{240} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{3}+\frac {1}{24} x^{5}-\frac {1}{112} x^{7}\right ) D\relax (y )\relax (0)+\frac {x^{3}}{6}-\frac {x^{5}}{24}+\frac {x^{7}}{112}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 77
AsymptoticDSolveValue[y''[x]+2*x*y'[x]-y[x]==x,y[x],{x,0,7}]
\[ y(x)\to \frac {x^7}{112}-\frac {x^5}{24}+\frac {x^3}{6}+c_2 \left (-\frac {x^7}{112}+\frac {x^5}{24}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {7 x^6}{240}-\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \]