Internal problem ID [6496]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (1-x^{2}\right ) y-\ln \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 52
Order:=6; dsolve(2*x^2*diff(y(x), x, x) - x*diff(y(x), x) + (-x^2 + 1)*y(x) = ln(x),y(x),type='series',x=1);
\[ y \relax (x ) = \left (1+\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}+\frac {37 \left (x -1\right )^{5}}{480}\right ) y \relax (1)+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{24}+\frac {19 \left (x -1\right )^{4}}{192}-\frac {119 \left (x -1\right )^{5}}{1920}\right ) D\relax (y )\relax (1)+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}+\frac {89 \left (x -1\right )^{5}}{960}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 105
AsymptoticDSolveValue[2*x^2*y''[x]-x*y'[x]+(1-x^2)*y[x]==Log[x],y[x],{x,1,5}]
\[ y(x)\to \frac {89}{960} (x-1)^5-\frac {3}{32} (x-1)^4+\frac {1}{12} (x-1)^3+c_1 \left (\frac {37}{480} (x-1)^5-\frac {5}{48} (x-1)^4+\frac {1}{6} (x-1)^3+1\right )+c_2 \left (-\frac {119 (x-1)^5}{1920}+\frac {19}{192} (x-1)^4-\frac {1}{24} (x-1)^3+\frac {1}{4} (x-1)^2+x-1\right ) \]