12.14.57 problem 68

Internal problem ID [1998]
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 : 68
Date solved : Tuesday, March 04, 2025 at 01:47:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

Order:=6; 
ode:=4*x^2*(x^2+2*x+3)*diff(diff(y(x),x),x)-x*(-15*x^2-14*x+3)*diff(y(x),x)+(7*x^2+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {2}{3} x +\frac {1}{9} x^{2}+\frac {4}{27} x^{3}-\frac {11}{81} x^{4}+\frac {10}{243} x^{5}\right ) \left (c_1 \,x^{{1}/{4}}+c_2 x \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 86

ode=4*x^2*(3+2*x+x^2)*D[y[x],{x,2}]-x*(3-14*x-15*x^2)*D[y[x],x]+(3+7*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (\frac {10 x^5}{243}-\frac {11 x^4}{81}+\frac {4 x^3}{27}+\frac {x^2}{9}-\frac {2 x}{3}+1\right )+c_2 \sqrt [4]{x} \left (\frac {10 x^5}{243}-\frac {11 x^4}{81}+\frac {4 x^3}{27}+\frac {x^2}{9}-\frac {2 x}{3}+1\right ) \]

✓ Sympy. Time used: 1.104 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x**2 + 2*x + 3)*Derivative(y(x), (x, 2)) - x*(-15*x**2 - 14*x + 3)*Derivative(y(x), x) + (7*x**2 + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x + C_{1} \sqrt [4]{x} + O\left (x^{6}\right ) \]