48.6.10 problem 10

Internal problem ID [9928]
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 : 10
Date solved : Tuesday, September 30, 2025 at 06:44:34 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 52

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

\[ y = c_1 \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\frac {27}{2240} x^{6}-\frac {81}{22400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\frac {243}{20} x^{6}-\frac {729}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 90

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

\[ y(x)\to c_1 \left (\frac {81 x^3}{80}+\frac {1}{x^3}-\frac {81 x^2}{40}-\frac {3}{x^2}+\frac {27 x}{8}+\frac {9}{2 x}-\frac {9}{2}\right )+c_2 \left (\frac {27 x^6}{2240}-\frac {81 x^5}{2240}+\frac {27 x^4}{280}-\frac {9 x^3}{40}+\frac {9 x^2}{20}-\frac {3 x}{4}+1\right ) \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (3*x + 4)*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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {81 x^{7}}{22400} + \frac {27 x^{6}}{2240} - \frac {81 x^{5}}{2240} + \frac {27 x^{4}}{280} - \frac {9 x^{3}}{40} + \frac {9 x^{2}}{20} - \frac {3 x}{4} + 1\right ) + \frac {C_{1} \left (\frac {9 x^{2}}{2} - 3 x + 1\right )}{x^{3}} + O\left (x^{8}\right ) \]