54.4.4 problem 4

Internal problem ID [8588]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 06:10:06 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 52

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

\[ y = c_{1} x^{{1}/{4}} \left (1-\frac {3}{5} x +\frac {1}{10} x^{2}-\frac {1}{130} x^{3}+\frac {3}{8840} x^{4}-\frac {3}{309400} x^{5}+\frac {3}{15470000} x^{6}-\frac {9}{3140410000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {3}{14} x^{2}-\frac {3}{154} x^{3}+\frac {3}{3080} x^{4}-\frac {9}{292600} x^{5}+\frac {9}{13459600} x^{6}-\frac {1}{94217200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {9 x^7}{3140410000}+\frac {3 x^6}{15470000}-\frac {3 x^5}{309400}+\frac {3 x^4}{8840}-\frac {x^3}{130}+\frac {x^2}{10}-\frac {3 x}{5}+1\right )+c_2 \left (-\frac {x^7}{94217200}+\frac {9 x^6}{13459600}-\frac {9 x^5}{292600}+\frac {3 x^4}{3080}-\frac {3 x^3}{154}+\frac {3 x^2}{14}-x+1\right ) \]

✓ Sympy. Time used: 0.933 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) + 3*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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{7}}{94217200} + \frac {9 x^{6}}{13459600} - \frac {9 x^{5}}{292600} + \frac {3 x^{4}}{3080} - \frac {3 x^{3}}{154} + \frac {3 x^{2}}{14} - x + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {3 x^{6}}{15470000} - \frac {3 x^{5}}{309400} + \frac {3 x^{4}}{8840} - \frac {x^{3}}{130} + \frac {x^{2}}{10} - \frac {3 x}{5} + 1\right ) + O\left (x^{8}\right ) \]