28.5.5 problem 10

Internal problem ID [7208]
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 : 10
Date solved : Tuesday, September 30, 2025 at 04:25:30 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 38

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {3}{2} x +\frac {3}{8} x^{2}+\frac {1}{16} x^{3}+\frac {3}{128} x^{4}+\frac {3}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-4 x +\frac {8}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 62

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

\[ y(x)\to c_2 \left (\frac {8 x^2}{3}-4 x+1\right )+c_1 \sqrt {x} \left (\frac {3 x^5}{256}+\frac {3 x^4}{128}+\frac {x^3}{16}+\frac {3 x^2}{8}-\frac {3 x}{2}+1\right ) \]

✓ Sympy. Time used: 0.400 (sec). Leaf size: 73

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

\[ y{\left (x \right )} = C_{2} \left (\frac {128 x^{5}}{14175} + \frac {32 x^{4}}{315} + \frac {32 x^{3}}{45} + \frac {8 x^{2}}{3} + 4 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {32 x^{4}}{2835} + \frac {32 x^{3}}{315} + \frac {8 x^{2}}{15} + \frac {4 x}{3} + 1\right ) + O\left (x^{6}\right ) \]