54.5.16 problem 16

Internal problem ID [8631]
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 : 16
Date solved : Sunday, March 30, 2025 at 01:21:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 46

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

\[ y = \frac {\left (\left (-15\right ) x -\frac {81}{4} x^{2}-\frac {3}{2} x^{3}+\frac {9}{32} x^{4}-\frac {27}{400} x^{5}+\frac {27}{1600} x^{6}-\frac {81}{19600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1+6 x +\frac {9}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 94

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

\[ y(x)\to \frac {c_1 \left (\frac {9 x^2}{2}+6 x+1\right )}{x}+c_2 \left (\frac {\left (\frac {9 x^2}{2}+6 x+1\right ) \log (x)}{x}+\frac {-\frac {81 x^7}{19600}+\frac {27 x^6}{1600}-\frac {27 x^5}{400}+\frac {9 x^4}{32}-\frac {3 x^3}{2}-\frac {81 x^2}{4}-15 x}{x}\right ) \]

✓ Sympy. Time used: 0.956 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {9 x^{2}}{2} + 6 x + 1\right )}{x} + O\left (x^{8}\right ) \]