54.5.15 problem 15

Internal problem ID [8630]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:21:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x \left (x +1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 56

Order:=8; 
ode:=4*x^2*diff(diff(y(x),x),x)+8*x*(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x -\frac {1}{4} x^{2}+\frac {1}{12} x^{3}-\frac {5}{192} x^{4}+\frac {7}{960} x^{5}-\frac {7}{3840} x^{6}+\frac {11}{26880} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-4\right ) x +\frac {3}{4} x^{2}-\frac {1}{4} x^{3}+\frac {31}{384} x^{4}-\frac {3}{128} x^{5}+\frac {419}{69120} x^{6}-\frac {97}{69120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 166

ode=4*x^2*D[y[x],{x,2}]+8*x*(x+1)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (\frac {11 x^7}{26880}-\frac {7 x^6}{3840}+\frac {7 x^5}{960}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {x^2}{4}+x+1\right )}{\sqrt {x}}+c_2 \left (\frac {-\frac {97 x^7}{69120}+\frac {419 x^6}{69120}-\frac {3 x^5}{128}+\frac {31 x^4}{384}-\frac {x^3}{4}+\frac {3 x^2}{4}-4 x}{\sqrt {x}}+\frac {\left (\frac {11 x^7}{26880}-\frac {7 x^6}{3840}+\frac {7 x^5}{960}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {x^2}{4}+x+1\right ) \log (x)}{\sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.890 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 8*x*(x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {11 x^{7}}{26880} - \frac {7 x^{6}}{3840} + \frac {7 x^{5}}{960} - \frac {5 x^{4}}{192} + \frac {x^{3}}{12} - \frac {x^{2}}{4} + x + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]