Internal problem ID [6530]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 59.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y-x \,{\mathrm e}^{x}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 53
Order:=6; dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=x*exp(x),y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) D\relax (y )\relax (0)+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 63
AsymptoticDSolveValue[(1-x^2)*y''[x]+y'[x]+y[x]==x*Exp[x],y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{120}-\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (\frac {7 x^5}{120}-\frac {x^4}{12}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]