Internal problem ID [5308]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number: 3(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (-2+4 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 55
Order:=8; dsolve(x^2*diff(y(x),x$2)-2*x^2*diff(y(x),x)+(4*x-2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{3} \left (1+\mathrm {O}\left (x^{8}\right )\right )+\ln \relax (x ) \left (\left (-48\right ) x^{3}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (12+36 x +72 x^{2}+88 x^{3}-24 x^{4}-\frac {24}{5} x^{5}-\frac {16}{15} x^{6}-\frac {8}{35} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 58
AsymptoticDSolveValue[x^2*y''[x]-2*x^2*y'[x]+(4*x-2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 x^2+c_1 \left (-4 x^2 \log (x)-\frac {4 x^6+18 x^5+90 x^4-390 x^3-270 x^2-135 x-45}{45 x}\right ) \]