87.26.32 problem 43

Internal problem ID [23866]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 43
Date solved : Thursday, October 02, 2025 at 09:46:01 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{1}/{4}} \left (1+\frac {9}{4} x +\frac {117}{32} x^{2}+\frac {663}{128} x^{3}+\frac {13923}{2048} x^{4}+\frac {69615}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {8}{3} x +\frac {32}{7} x^{2}+\frac {512}{77} x^{3}+\frac {2048}{231} x^{4}+\frac {16384}{1463} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {69615 x^5}{8192}+\frac {13923 x^4}{2048}+\frac {663 x^3}{128}+\frac {117 x^2}{32}+\frac {9 x}{4}+1\right )+c_2 \left (\frac {16384 x^5}{1463}+\frac {2048 x^4}{231}+\frac {512 x^3}{77}+\frac {32 x^2}{7}+\frac {8 x}{3}+1\right ) \]

✓ Sympy. Time used: 0.451 (sec). Leaf size: 75

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {4096 x^{5}}{987525} + \frac {512 x^{4}}{10395} - \frac {256 x^{3}}{693} + \frac {32 x^{2}}{21} - \frac {8 x}{3} + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {512 x^{4}}{29835} - \frac {256 x^{3}}{1755} + \frac {32 x^{2}}{45} - \frac {8 x}{5} + 1\right ) + O\left (x^{6}\right ) \]