54.7.2 problem 2

Internal problem ID [8650]
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 : 2
Date solved : Sunday, March 30, 2025 at 01:22:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 72

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

\[ y = x \left (c_1 \,x^{2} \left (1-\frac {4}{3} x +\frac {2}{3} x^{2}-\frac {8}{45} x^{3}+\frac {4}{135} x^{4}-\frac {16}{4725} x^{5}+\frac {4}{14175} x^{6}-\frac {16}{893025} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\left (16 x^{2}-\frac {64}{3} x^{3}+\frac {32}{3} x^{4}-\frac {128}{45} x^{5}+\frac {64}{135} x^{6}-\frac {256}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right )+\left (-2-8 x +\frac {256}{9} x^{3}-\frac {200}{9} x^{4}+\frac {5024}{675} x^{5}-\frac {2912}{2025} x^{6}+\frac {90752}{496125} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 121

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

\[ y(x)\to c_1 \left (\frac {x \left (1696 x^6-8976 x^5+27900 x^4-39600 x^3+8100 x^2+8100 x+2025\right )}{2025}-\frac {8}{135} x^3 \left (4 x^4-24 x^3+90 x^2-180 x+135\right ) \log (x)\right )+c_2 \left (\frac {4 x^9}{14175}-\frac {16 x^8}{4725}+\frac {4 x^7}{135}-\frac {8 x^6}{45}+\frac {2 x^5}{3}-\frac {4 x^4}{3}+x^3\right ) \]

✓ Sympy. Time used: 0.822 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{1} x^{3} \left (\frac {4 x^{4}}{135} - \frac {8 x^{3}}{45} + \frac {2 x^{2}}{3} - \frac {4 x}{3} + 1\right ) + O\left (x^{8}\right ) \]