12.14.38 problem 40

Internal problem ID [1979]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 40
Date solved : Saturday, March 29, 2025 at 11:44:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 34

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

\[ y = \frac {c_1 x \left (1-\frac {1}{4} x^{2}+\frac {7}{80} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{8} x^{2}+\frac {5}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (\frac {5 x^{7/2}}{128}-\frac {x^{3/2}}{8}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {7 x^{9/2}}{80}-\frac {x^{5/2}}{4}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 1.133 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + x*(7*x**2 + 4)*Derivative(y(x), x) - (1 - 3*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + \frac {C_{1}}{\sqrt {x}} + O\left (x^{6}\right ) \]