12.14.10 problem 7

Internal problem ID [1951]
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 : 7
Date solved : Tuesday, March 04, 2025 at 01:46:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 8 x^{2} y^{\prime \prime }-2 x \left (-x^{2}-4 x +3\right ) y^{\prime }+\left (x^{2}+6 x +3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 47

Order:=6; 
ode:=8*x^2*diff(diff(y(x),x),x)-2*x*(-x^2-4*x+3)*diff(y(x),x)+(x^2+6*x+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1+4 x -\frac {131}{24} x^{2}+\frac {39}{14} x^{3}-\frac {19865}{29568} x^{4}+\frac {4421}{110880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{{3}/{2}} \left (1-x +\frac {11}{26} x^{2}-\frac {109}{1326} x^{3}+\frac {5}{12376} x^{4}+\frac {229}{71400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 86

ode=8*x^2*D[y[x],{x,2}]-2*x*(3-4*x-x^2)*D[y[x],x]+(3+6*x+x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {4421 x^5}{110880}-\frac {19865 x^4}{29568}+\frac {39 x^3}{14}-\frac {131 x^2}{24}+4 x+1\right ) \sqrt [4]{x}+c_1 \left (\frac {229 x^5}{71400}+\frac {5 x^4}{12376}-\frac {109 x^3}{1326}+\frac {11 x^2}{26}-x+1\right ) x^{3/2} \]

✓ Sympy. Time used: 1.169 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*Derivative(y(x), (x, 2)) - 2*x*(-x**2 - 4*x + 3)*Derivative(y(x), x) + (x**2 + 6*x + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (- \frac {109 x^{3}}{1326} + \frac {11 x^{2}}{26} - x + 1\right ) + C_{1} \sqrt [4]{x} \left (- \frac {19865 x^{4}}{29568} + \frac {39 x^{3}}{14} - \frac {131 x^{2}}{24} + 4 x + 1\right ) + O\left (x^{6}\right ) \]