28.5.1 problem 5

Internal problem ID [7204]
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 : 5
Date solved : Tuesday, September 30, 2025 at 04:25:26 PM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 34

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

\[ y = \frac {c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 \left (1+\frac {1}{6} x +\frac {3}{40} x^{2}+\frac {5}{112} x^{3}+\frac {35}{1152} x^{4}+\frac {63}{2816} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \left (\frac {63 x^5}{2816}+\frac {35 x^4}{1152}+\frac {5 x^3}{112}+\frac {3 x^2}{40}+\frac {x}{6}+1\right )+\frac {c_2}{\sqrt {x}} \]

✓ Sympy. Time used: 0.501 (sec). Leaf size: 14

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

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