Internal problem ID [5723]
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(a).
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-x^{2}=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: 38
Order:=8; dsolve(diff(y(x),x$2)+2*x*y(x)=x^2,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{3} x^{3}+\frac {1}{45} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{4}+\frac {1}{126} x^{7}\right ) D\relax (y )\relax (0)+\frac {x^{4}}{12}-\frac {x^{7}}{252}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 56
AsymptoticDSolveValue[y''[x]+2*x*y[x]==x^2,y[x],{x,0,7}]
\[ y(x)\to -\frac {x^7}{252}+\frac {x^4}{12}+c_2 \left (\frac {x^7}{126}-\frac {x^4}{6}+x\right )+c_1 \left (\frac {x^6}{45}-\frac {x^3}{3}+1\right ) \]