54.7.12 problem 12

Internal problem ID [8660]
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 : 12
Date solved : Wednesday, March 05, 2025 at 06:11:47 AM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 76

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

\[ y = \frac {c_{1} \left (1+\frac {5}{3} x +\frac {5}{4} x^{2}+\frac {7}{12} x^{3}+\frac {7}{36} x^{4}+\frac {1}{20} x^{5}+\frac {1}{96} x^{6}+\frac {11}{6048} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (12 x^{2}+20 x^{3}+15 x^{4}+7 x^{5}+\frac {7}{3} x^{6}+\frac {3}{5} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+6 x +7 x^{2}-11 x^{3}-17 x^{4}-\frac {32}{3} x^{5}-\frac {305}{72} x^{6}-\frac {737}{600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.1 (sec). Leaf size: 116

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

\[ y(x)\to c_2 \left (\frac {x^6}{96}+\frac {x^5}{20}+\frac {7 x^4}{36}+\frac {7 x^3}{12}+\frac {5 x^2}{4}+\frac {5 x}{3}+1\right )+c_1 \left (\frac {389 x^6+1020 x^5+1764 x^4+1512 x^3-72 x^2-432 x+144}{144 x^2}-\frac {1}{6} \left (7 x^4+21 x^3+45 x^2+60 x+36\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.840 (sec). Leaf size: 51

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

\[ y{\left (x \right )} = C_{1} \left (\frac {11 x^{7}}{6048} + \frac {x^{6}}{96} + \frac {x^{5}}{20} + \frac {7 x^{4}}{36} + \frac {7 x^{3}}{12} + \frac {5 x^{2}}{4} + \frac {5 x}{3} + 1\right ) + O\left (x^{8}\right ) \]