Internal problem ID [4222]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XVI. page 220
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 34
Order:=6; dsolve(4*diff(y(x),x$2)+3*(2-x^2)/(1-x^2)^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {3}{4} x^{2}-\frac {3}{32} x^{4}\right ) y \relax (0)+\left (x -\frac {1}{4} x^{3}-\frac {3}{32} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 42
AsymptoticDSolveValue[4*y''[x]+3*(2-x^2)/(1-x^2)^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {3 x^5}{32}-\frac {x^3}{4}+x\right )+c_1 \left (-\frac {3 x^4}{32}-\frac {3 x^2}{4}+1\right ) \]