74.17.12 problem 12

Internal problem ID [16464]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 12
Date solved : Thursday, March 13, 2025 at 08:14:18 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+\left (\frac {16}{3 x}-1\right ) y^{\prime }-\frac {16 y}{3 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.051 (sec). Leaf size: 45

Order:=6; 
ode:=diff(diff(y(x),x),x)+(16/3/x-1)*diff(y(x),x)-16/3/x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{16}/{3}}}+c_{2} x \left (1+\frac {3}{22} x +\frac {9}{550} x^{2}+\frac {27}{15400} x^{3}+\frac {81}{477400} x^{4}+\frac {243}{16231600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 82

ode=D[y[x],{x,2}]+(16/3*1/x-1)*D[y[x],x]-16/3*1/x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (\frac {243 x^5}{16231600}+\frac {81 x^4}{477400}+\frac {27 x^3}{15400}+\frac {9 x^2}{550}+\frac {3 x}{22}+1\right )+\frac {c_2 \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )}{x^{16/3}} \]

✓ Sympy. Time used: 1.114 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-1 + 16/(3*x))*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 16*y(x)/(3*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {81 x^{4}}{477400} + \frac {27 x^{3}}{15400} + \frac {9 x^{2}}{550} + \frac {3 x}{22} + 1\right ) + \frac {C_{1} \left (\frac {x^{10}}{3628800} + \frac {x^{9}}{362880} + \frac {x^{8}}{40320} + \frac {x^{7}}{5040} + \frac {x^{6}}{720} + \frac {x^{5}}{120} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right )}{x^{\frac {16}{3}}} + O\left (x^{6}\right ) \]