20.26.7 problem (d)

Internal problem ID [4032]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : (d)
Date solved : Tuesday, March 04, 2025 at 05:23:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (4 x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}\right ) y^{\prime }-\frac {7 y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 60

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(4*x+1/2*x^2-1/3*x^3)*diff(y(x),x)-7/4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{20} x +\frac {49}{2880} x^{2}-\frac {533}{241920} x^{3}+\frac {277}{491520} x^{4}-\frac {203759}{2388787200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {8491}{768} x^{4}-\frac {8491}{15360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+84 x -\frac {273}{4} x^{2}+\frac {1939}{48} x^{3}-\frac {221}{12} x^{4}-\frac {49993}{57600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{7}/{2}}} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 93

ode=x^2*D[y[x],{x,2}]+(4*x+1/2*x^2-1/3*x^3)*D[y[x],x]-7/4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {277 x^{9/2}}{491520}-\frac {533 x^{7/2}}{241920}+\frac {49 x^{5/2}}{2880}-\frac {x^{3/2}}{20}+\sqrt {x}\right )+c_1 \left (\frac {65067 x^4-124096 x^3+209664 x^2-258048 x+442368}{442368 x^{7/2}}-\frac {8491 \sqrt {x} \log (x)}{110592}\right ) \]

✓ Sympy. Time used: 1.180 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (-x**3/3 + x**2/2 + 4*x)*Derivative(y(x), x) - 7*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {277 x^{4}}{491520} - \frac {533 x^{3}}{241920} + \frac {49 x^{2}}{2880} - \frac {x}{20} + 1\right ) + O\left (x^{6}\right ) \]