6.15.52 problem 48

Internal problem ID [2050]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 48
Date solved : Tuesday, September 30, 2025 at 05:23:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*(1-x)*diff(diff(y(x),x),x)-x*(3-5*x)*diff(y(x),x)+(4-5*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\left (4 x -7 x^{2}+\frac {11}{3} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-3 x +3 x^{2}-x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \left (-x^3+3 x^2-3 x+1\right ) x^2+c_2 \left (\left (-x^3+3 x^2-3 x+1\right ) x^2 \log (x)+\left (-\frac {x^5}{20}-\frac {x^4}{4}+\frac {11 x^3}{3}-7 x^2+4 x\right ) x^2\right ) \]

✓ Sympy. Time used: 0.421 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) - x*(3 - 5*x)*Derivative(y(x), x) + (4 - 5*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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