15.23.14 problem 18

Internal problem ID [3364]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 18
Date solved : Tuesday, March 04, 2025 at 04:36:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 42

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

\[ y \left (x \right ) = \frac {c_{2} x^{{5}/{2}} \left (1-\frac {3}{14} x -\frac {3}{56} x^{2}-\frac {45}{1232} x^{3}-\frac {675}{18304} x^{4}-\frac {1701}{36608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-12 x +72 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 63

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

\[ y(x)\to \frac {c_2 \left (72 x^2-12 x+1\right )}{x^2}+c_1 \sqrt {x} \left (-\frac {1701 x^5}{36608}-\frac {675 x^4}{18304}-\frac {45 x^3}{1232}-\frac {3 x^2}{56}-\frac {3 x}{14}+1\right ) \]

✓ Sympy. Time used: 0.999 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(1 - 3*x)*Derivative(y(x), (x, 2)) + 5*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)
 

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