48.7.11 problem 11

Internal problem ID [9945]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 06:44:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-5 x y^{\prime }+\left (8+5 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 74

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)-5*x*diff(y(x),x)+(8+5*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_1 \,x^{2} \left (1-\frac {5}{3} x +\frac {25}{24} x^{2}-\frac {25}{72} x^{3}+\frac {125}{1728} x^{4}-\frac {125}{12096} x^{5}+\frac {625}{580608} x^{6}-\frac {3125}{36578304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\left (25 x^{2}-\frac {125}{3} x^{3}+\frac {625}{24} x^{4}-\frac {625}{72} x^{5}+\frac {3125}{1728} x^{6}-\frac {3125}{12096} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (-2-10 x +\frac {500}{9} x^{3}-\frac {15625}{288} x^{4}+\frac {19625}{864} x^{5}-\frac {56875}{10368} x^{6}+\frac {443125}{508032} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) x^{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 123

ode=x^2*D[y[x],{x,2}]-5*x*D[y[x],x]+(8+5*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^2 \left (33125 x^6-140250 x^5+348750 x^4-396000 x^3+64800 x^2+51840 x+10368\right )}{10368}-\frac {25 x^4 \left (125 x^4-600 x^3+1800 x^2-2880 x+1728\right ) \log (x)}{3456}\right )+c_2 \left (\frac {625 x^{10}}{580608}-\frac {125 x^9}{12096}+\frac {125 x^8}{1728}-\frac {25 x^7}{72}+\frac {25 x^6}{24}-\frac {5 x^5}{3}+x^4\right ) \]

✓ Sympy. Time used: 0.261 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} x^{4} \left (- \frac {25 x^{3}}{72} + \frac {25 x^{2}}{24} - \frac {5 x}{3} + 1\right ) + O\left (x^{8}\right ) \]