Internal problem ID [6152]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page
355
Problem number: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+x y^{\prime }+3 y-x^{2}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 48
Order:=8; dsolve(diff(y(x),x$2)+x*diff(y(x),x)+3*y(x)=x^2,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {3}{2} x^{2}+\frac {5}{8} x^{4}-\frac {7}{48} x^{6}\right ) y \relax (0)+\left (x -\frac {2}{3} x^{3}+\frac {1}{5} x^{5}-\frac {4}{105} x^{7}\right ) D\relax (y )\relax (0)+\frac {x^{4}}{12}-\frac {7 x^{6}}{360}+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 70
AsymptoticDSolveValue[y''[x]+x*y'[x]+3*y[x]==x^2,y[x],{x,0,7}]
\[ y(x)\to -\frac {7 x^6}{360}+\frac {x^4}{12}+c_2 \left (-\frac {4 x^7}{105}+\frac {x^5}{5}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {7 x^6}{48}+\frac {5 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]