54.5.5 problem 5

Internal problem ID [8620]
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 : 5
Date solved : Sunday, March 30, 2025 at 01:21:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+\left (1+5 x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 52

Order:=8; 
ode:=x*(1+x)*diff(diff(y(x),x),x)+(1+5*x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (2 x -\frac {11}{2} x^{2}+\frac {21}{2} x^{3}-17 x^{4}+25 x^{5}-\frac {69}{2} x^{6}+\frac {91}{2} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+28 x^{6}-36 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 125

ode=x*(1+x)*D[y[x],{x,2}]+(1+5*x)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-36 x^7+28 x^6-21 x^5+15 x^4-10 x^3+6 x^2-3 x+1\right )+c_2 \left (\frac {91 x^7}{2}-\frac {69 x^6}{2}+25 x^5-17 x^4+\frac {21 x^3}{2}-\frac {11 x^2}{2}+\left (-36 x^7+28 x^6-21 x^5+15 x^4-10 x^3+6 x^2-3 x+1\right ) \log (x)+2 x\right ) \]

✓ Sympy. Time used: 1.006 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) + (5*x + 1)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {27 x^{7}}{313600} + \frac {9 x^{6}}{6400} - \frac {27 x^{5}}{1600} + \frac {9 x^{4}}{64} - \frac {3 x^{3}}{4} + \frac {9 x^{2}}{4} - 3 x + 1\right ) + O\left (x^{8}\right ) \]