6.14.37 problem 39

Internal problem ID [1978]
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 : 39
Date solved : Tuesday, September 30, 2025 at 05:22:03 AM
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}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

Order:=6; 
ode:=2*x^2*(x^2+2)*diff(diff(y(x),x),x)-x*(-7*x^2+12)*diff(y(x),x)+(3*x^2+7)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (x^{3} \left (1-\frac {9}{8} x^{2}+\frac {117}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +\left (12+9 x^{2}-\frac {63}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 58

ode=2*x^2*(2+x^2)*D[y[x],{x,2}]-x*(12-7*x^2)*D[y[x],x]+(7+3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {21 x^{9/2}}{16}+\frac {3 x^{5/2}}{4}+\sqrt {x}\right )+c_2 \left (\frac {117 x^{15/2}}{128}-\frac {9 x^{11/2}}{8}+x^{7/2}\right ) \]

✓ Sympy. Time used: 0.510 (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*(12 - 7*x**2)*Derivative(y(x), x) + (3*x**2 + 7)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {7}{2}} + C_{1} \sqrt {x} + O\left (x^{6}\right ) \]