54.3.24 problem 24

Internal problem ID [8580]
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 : 24
Date solved : Wednesday, March 05, 2025 at 06:09:57 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (3 x^{2}+1\right ) y^{\prime \prime }+13 x y^{\prime }+7 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

Order:=8; 
ode:=(3*x^2+1)*diff(diff(y(x),x),x)+13*x*diff(y(x),x)+7*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {7}{2} x^{2}+\frac {91}{8} x^{4}-\frac {1729}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {10}{3} x^{3}+\frac {32}{3} x^{5}-\frac {704}{21} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 56

ode=(1+3*x^2)*D[y[x],{x,2}]+13*x*D[y[x],x]+7*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {704 x^7}{21}+\frac {32 x^5}{3}-\frac {10 x^3}{3}+x\right )+c_1 \left (-\frac {1729 x^6}{48}+\frac {91 x^4}{8}-\frac {7 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.860 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*x*Derivative(y(x), x) + (3*x**2 + 1)*Derivative(y(x), (x, 2)) + 7*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {1729 x^{6}}{48} + \frac {91 x^{4}}{8} - \frac {7 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {32 x^{4}}{3} - \frac {10 x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]