28.5.6 problem 11

Internal problem ID [7209]
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 : Tuesday, September 30, 2025 at 04:25:31 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*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 39

Order:=6; 
ode:=4*diff(diff(y(x),x),x)+3*(-x^2+2)/(-x^2+1)^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=4*D[y[x],{x,2}]+3*(2-x^2)/(1-x^2)^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.348 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*Derivative(y(x), (x, 2)) + (6 - 3*x**2)*y(x)/(1 - x**2)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {21 x^{5} r{\left (3 \right )}}{40} + C_{2} \left (- \frac {3 x^{4}}{32} - \frac {3 x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {3 x^{4}}{80} + 1\right ) + O\left (x^{6}\right ) \]