54.4.1 problem 1

Internal problem ID [8585]
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 : 1
Date solved : Sunday, March 30, 2025 at 01:20:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 40

Order:=8; 
ode:=2*x*(1+x)*diff(diff(y(x),x),x)+3*(1+x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+x +\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_2 \left (1+\frac {1}{3} x -\frac {1}{15} x^{2}+\frac {1}{35} x^{3}-\frac {1}{63} x^{4}+\frac {1}{99} x^{5}-\frac {1}{143} x^{6}+\frac {1}{195} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 67

ode=2*x*(x+1)*D[y[x],{x,2}]+3*(x+1)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^7}{195}-\frac {x^6}{143}+\frac {x^5}{99}-\frac {x^4}{63}+\frac {x^3}{35}-\frac {x^2}{15}+\frac {x}{3}+1\right )+\frac {c_2 (x+1)}{\sqrt {x}} \]

✓ Sympy. Time used: 1.113 (sec). Leaf size: 107

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(x + 1)*Derivative(y(x), (x, 2)) + (3*x + 3)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {8 x^{7}}{638512875} + \frac {4 x^{6}}{6081075} + \frac {4 x^{5}}{155925} + \frac {2 x^{4}}{2835} + \frac {4 x^{3}}{315} + \frac {2 x^{2}}{15} + \frac {2 x}{3} + 1\right ) + \frac {C_{1} \left (\frac {8 x^{7}}{42567525} + \frac {4 x^{6}}{467775} + \frac {4 x^{5}}{14175} + \frac {2 x^{4}}{315} + \frac {4 x^{3}}{45} + \frac {2 x^{2}}{3} + 2 x + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]