15.23.15 problem 19

Internal problem ID [3365]
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 : 19
Date solved : Sunday, March 30, 2025 at 01:38:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+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: 0.025 (sec). Leaf size: 47

Order:=6; 
ode:=4*x^2*(1+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);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{4} x +\frac {5}{32} x^{2}-\frac {15}{128} x^{3}+\frac {195}{2048} x^{4}-\frac {663}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{2} \left (1-\frac {8}{11} x +\frac {32}{55} x^{2}-\frac {512}{1045} x^{3}+\frac {2048}{4807} x^{4}-\frac {16384}{43263} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 88

ode=4*x^2*(1+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}]
 

\[ y(x)\to c_1 \left (-\frac {16384 x^5}{43263}+\frac {2048 x^4}{4807}-\frac {512 x^3}{1045}+\frac {32 x^2}{55}-\frac {8 x}{11}+1\right ) x^2+c_2 \left (-\frac {663 x^5}{8192}+\frac {195 x^4}{2048}-\frac {15 x^3}{128}+\frac {5 x^2}{32}-\frac {x}{4}+1\right ) \sqrt [4]{x} \]

✓ Sympy. Time used: 0.973 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x + 1)*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)
 

\[ y{\left (x \right )} = C_{2} x^{2} + C_{1} \sqrt [4]{x} + O\left (x^{6}\right ) \]