54.3.22 problem 22

Internal problem ID [8578]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 22
Date solved : Wednesday, March 05, 2025 at 06:09:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+4\right ) y^{\prime \prime }+3 x y^{\prime }-8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {5}{24} x^{3}-\frac {7}{384} x^{5}+\frac {3}{1024} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode=(x^2+4)*D[y[x],{x,2}]+3*x*D[y[x],x]-8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (x^2+1\right )+c_2 \left (\frac {3 x^7}{1024}-\frac {7 x^5}{384}+\frac {5 x^3}{24}+x\right ) \]

✓ Sympy. Time used: 0.853 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (x^{2} + 1\right ) + C_{1} x \left (- \frac {7 x^{4}}{384} + \frac {5 x^{2}}{24} + 1\right ) + O\left (x^{8}\right ) \]