Internal problem ID [6499]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 27.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 35
Order:=6; dsolve(x^2*(2-x^2)*diff(y(x), x$2) - x*(3+4*x^2)*diff(y(x), x) + (2-2*x^2)*y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {15}{8} x^{2}+\frac {189}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {6}{7} x^{2}+\frac {45}{77} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 50
AsymptoticDSolveValue[x^2*(2-x^2)*y''[x] - x*(3+4*x^2)*y'[x] + (2-2*x^2)*y[x] == 0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {45 x^4}{77}+\frac {6 x^2}{7}+1\right ) x^2+c_2 \left (\frac {189 x^4}{128}+\frac {15 x^2}{8}+1\right ) \sqrt {x} \]