12.14.7 problem 4

Internal problem ID [1948]
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 : 4
Date solved : Saturday, March 29, 2025 at 11:43:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

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

\[ y = \frac {c_2 \,x^{{5}/{4}} \left (1-\frac {1}{2} x -\frac {19}{104} x^{2}+\frac {1571}{10608} x^{3}+\frac {3225}{198016} x^{4}-\frac {752183}{29702400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+2 x -\frac {11}{6} x^{2}-\frac {1}{7} x^{3}+\frac {895}{1848} x^{4}-\frac {499}{13860} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

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

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

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {752183 x^5}{29702400}+\frac {3225 x^4}{198016}+\frac {1571 x^3}{10608}-\frac {19 x^2}{104}-\frac {x}{2}+1\right )+\frac {c_2 \left (-\frac {499 x^5}{13860}+\frac {895 x^4}{1848}-\frac {x^3}{7}-\frac {11 x^2}{6}+2 x+1\right )}{x} \]

✓ Sympy. Time used: 1.108 (sec). Leaf size: 78

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

\[ y{\left (x \right )} = C_{2} \sqrt [4]{x} \left (\frac {3225 x^{4}}{198016} + \frac {1571 x^{3}}{10608} - \frac {19 x^{2}}{104} - \frac {x}{2} + 1\right ) + \frac {C_{1} \left (- \frac {27011 x^{6}}{351120} - \frac {499 x^{5}}{13860} + \frac {895 x^{4}}{1848} - \frac {x^{3}}{7} - \frac {11 x^{2}}{6} + 2 x + 1\right )}{x} + O\left (x^{6}\right ) \]