50.19.14 problem 4(a)

Internal problem ID [8122]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 4(a)
Date solved : Sunday, March 30, 2025 at 12:45:59 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x y^{\prime \prime }+3 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 52

Order:=8; 
ode:=4*x*diff(diff(y(x),x),x)+3*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{5} x +\frac {1}{90} x^{2}-\frac {1}{3510} x^{3}+\frac {1}{238680} x^{4}-\frac {1}{25061400} x^{5}+\frac {1}{3759210000} x^{6}-\frac {1}{763119630000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-\frac {1}{3} x +\frac {1}{42} x^{2}-\frac {1}{1386} x^{3}+\frac {1}{83160} x^{4}-\frac {1}{7900200} x^{5}+\frac {1}{1090227600} x^{6}-\frac {1}{206053016400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 113

ode=4*x*D[y[x],{x,2}]+3*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {x^7}{763119630000}+\frac {x^6}{3759210000}-\frac {x^5}{25061400}+\frac {x^4}{238680}-\frac {x^3}{3510}+\frac {x^2}{90}-\frac {x}{5}+1\right )+c_2 \left (-\frac {x^7}{206053016400}+\frac {x^6}{1090227600}-\frac {x^5}{7900200}+\frac {x^4}{83160}-\frac {x^3}{1386}+\frac {x^2}{42}-\frac {x}{3}+1\right ) \]

✓ Sympy. Time used: 0.953 (sec). Leaf size: 80

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) + y(x) + 3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{7}}{206053016400} + \frac {x^{6}}{1090227600} - \frac {x^{5}}{7900200} + \frac {x^{4}}{83160} - \frac {x^{3}}{1386} + \frac {x^{2}}{42} - \frac {x}{3} + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {x^{6}}{3759210000} - \frac {x^{5}}{25061400} + \frac {x^{4}}{238680} - \frac {x^{3}}{3510} + \frac {x^{2}}{90} - \frac {x}{5} + 1\right ) + O\left (x^{8}\right ) \]