50.19.12 problem 3(c)

Internal problem ID [8120]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 3(c)
Date solved : Sunday, March 30, 2025 at 12:45:55 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 74

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 116

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

\[ y(x)\to c_2 \left (\frac {4 x^6}{14175}-\frac {16 x^5}{4725}+\frac {4 x^4}{135}-\frac {8 x^3}{45}+\frac {2 x^2}{3}-\frac {4 x}{3}+1\right )+c_1 \left (\frac {1696 x^6-8976 x^5+27900 x^4-39600 x^3+8100 x^2+8100 x+2025}{2025 x^2}-\frac {8}{135} \left (4 x^4-24 x^3+90 x^2-180 x+135\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.879 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {16 x^{7}}{893025} + \frac {4 x^{6}}{14175} - \frac {16 x^{5}}{4725} + \frac {4 x^{4}}{135} - \frac {8 x^{3}}{45} + \frac {2 x^{2}}{3} - \frac {4 x}{3} + 1\right ) + O\left (x^{8}\right ) \]