34.5.6 problem 11

Internal problem ID [6074]
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
Date solved : Monday, January 27, 2025 at 01:34:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

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);
 

\[ y = \left (1-\frac {3}{4} x^{2}-\frac {3}{32} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}-\frac {3}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 42

AsymptoticDSolveValue[4*D[y[x],{x,2}]+3*(2-x^2)/(1-x^2)^2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {3 x^5}{32}-\frac {x^3}{4}+x\right )+c_1 \left (-\frac {3 x^4}{32}-\frac {3 x^2}{4}+1\right ) \]