48.6.12 problem 12

Internal problem ID [9930]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:44:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (3+2 x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 50

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(2*x+3)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {4}{3} x +x^{2}-\frac {8}{15} x^{3}+\frac {2}{9} x^{4}-\frac {8}{105} x^{5}+\frac {1}{45} x^{6}-\frac {16}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (-2+4 x^{2}-\frac {16}{3} x^{3}+4 x^{4}-\frac {32}{15} x^{5}+\frac {8}{9} x^{6}-\frac {32}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 77

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

\[ y(x)\to c_1 \left (-\frac {4 x^4}{9}+\frac {16 x^3}{15}-2 x^2+\frac {1}{x^2}+\frac {8 x}{3}-2\right )+c_2 \left (\frac {x^6}{45}-\frac {8 x^5}{105}+\frac {2 x^4}{9}-\frac {8 x^3}{15}+x^2-\frac {4 x}{3}+1\right ) \]

✓ Sympy. Time used: 0.277 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (2*x + 3)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {16 x^{7}}{2835} + \frac {x^{6}}{45} - \frac {8 x^{5}}{105} + \frac {2 x^{4}}{9} - \frac {8 x^{3}}{15} + x^{2} - \frac {4 x}{3} + 1\right ) + \frac {C_{1}}{x^{2}} + O\left (x^{8}\right ) \]