7.15.23 problem 23

Internal problem ID [479]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 23
Date solved : Saturday, March 29, 2025 at 04:54:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+7 x y^{\prime }-\left (x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 36

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

\[ y = \frac {c_2 \,x^{{7}/{6}} \left (1+\frac {1}{38} x^{2}+\frac {1}{4712} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {1}{10} x^{2}+\frac {1}{680} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{3}}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 52

ode=6*x^2*D[y[x],{x,2}]+7*x*D[y[x],x]-(x^2+2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{4712}+\frac {x^2}{38}+1\right )+\frac {c_2 \left (\frac {x^4}{680}+\frac {x^2}{10}+1\right )}{x^{2/3}} \]

✓ Sympy. Time used: 0.939 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {x^{4}}{4712} + \frac {x^{2}}{38} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{680} + \frac {x^{2}}{10} + 1\right )}{x^{\frac {2}{3}}} + O\left (x^{6}\right ) \]