34.4.11 problem 11

Internal problem ID [6065]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 11
Date solved : Sunday, March 30, 2025 at 10:37:35 AM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 60

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

\[ y = c_1 \,x^{2} \left (1+\frac {3}{4} x +\frac {75}{128} x^{2}+\frac {245}{512} x^{3}+\frac {6615}{16384} x^{4}+\frac {22869}{65536} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\frac {1}{16} x^{2}+\frac {3}{64} x^{3}+\frac {75}{2048} x^{4}+\frac {245}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x +\frac {1}{2} x^{2}+\frac {3}{8} x^{3}+\frac {2415}{8192} x^{4}+\frac {23779}{98304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 86

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

\[ y(x)\to c_1 \left (\frac {135 x^4+192 x^3+256 x^2-4096 x+16384}{16384}-\frac {x^2 \left (75 x^2+96 x+128\right ) \log (x)}{4096}\right )+c_2 \left (\frac {6615 x^6}{16384}+\frac {245 x^5}{512}+\frac {75 x^4}{128}+\frac {3 x^3}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.950 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = C_{2} x^{2} + C_{1} + O\left (x^{6}\right ) \]